Chapter 4 Resource Masters
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CONSUMABLE WORKBOOKS Many of the worksheets contained in the Chapter Resource Masters
booklets are available as consumable workbooks in both English and Spanish.
MHID
ISBN
Study Guide and Intervention Workbook
0-07-660292-3
978-0-07-660292-6
Homework Practice Workbook
0-07-660291-5 978-0-07-660291-9
Spanish Version
Homework Practice Workbook
0-07-660294-X
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Answers For Workbooks The answers for Chapter 4 of these workbooks can be found in the
back of this Chapter Resource Masters booklet.
ConnectED All of the materials found in this booklet are included for viewing, printing, and editing at
connected.mcgraw-hill.com.
Spanish Assessment Masters (MHID: 0-07-660289-3, ISBN: 978-0-07-660289-6) These masters
contain a Spanish version of Chapter 4 Test Form 2A and Form 2C.
iii
Teacher’s Guide to Using the Chapter 4
Resource Masters ..............................................iv
Chapter Resources
Chapter 4 Student-Built Glossary ...................... 1
Chapter 4 Anticipation Guide (English) ............. 3
Chapter 4 Anticipation Guide (Spanish) ............ 4
Lesson 4-1
Graphing Equations in Slope-Intercept Form
Study Guide and Intervention ............................ 5
Skills Practice .................................................... 7
Practice ............................................................. 8
Word Problem Practice ...................................... 9
Enrichment ...................................................... 10
Lesson 4-2
Writing Equations in Slope-Intercept Form
Study Guide and Intervention ...........................11
Skills Practice .................................................. 13
Practice ........................................................... 14
Word Problem Practice .................................... 15
Enrichment ...................................................... 16
Lesson 4-3
Writing Equations in Point-Slope Form
Study Guide and Intervention .......................... 17
Skills Practice .................................................. 19
Practice ........................................................... 20
Word Problem Practice .................................... 21
Enrichment ...................................................... 22
Graphing Calculator Activity ............................ 23
Lesson 4-4
Parallel and Perpendicular Lines
Study Guide and Intervention .......................... 24
Skills Practice .................................................. 26
Practice ........................................................... 27
Word Problem Practice .................................... 28
Enrichment ...................................................... 29
Lesson 4-5
Scatter Plots and Lines of Fit
Study Guide and Intervention .......................... 30
Skills Practice .................................................. 32
Practice ........................................................... 33
Word Problem Practice .................................... 34
Enrichment ...................................................... 35
Spreadsheet Activity ........................................ 36
Lesson 4-6
Regression and Median-Fit Lines
Study Guide and Intervention .......................... 37
Skills Practice .................................................. 39
Practice ........................................................... 40
Word Problem Practice .................................... 41
Enrichment ...................................................... 42
Lesson 4-7
Inverse Linear Functions
Study Guide and Intervention .......................... 43
Skills Practice .................................................. 45
Practice ........................................................... 46
Word Problem Practice .................................... 47
Enrichment ...................................................... 48
Assessment
Student Recording Sheet ................................ 49
Rubric for Scoring Extended Response .......... 50
Chapter 4 Quizzes 1 and 2 ............................. 51
Chapter 4 Quizzes 3 and 4 ............................. 52
Chapter 4 Mid-Chapter Test ............................ 53
Chapter 4 Vocabulary Test ............................... 54
Chapter 4 Test, Form 1 .................................... 55
Chapter 4 Test, Form 2A ................................. 57
Chapter 4 Test, Form 2B ................................. 59
Chapter 4 Test Form 2C .................................. 61
Chapter 4 Test Form 2D .................................. 63
Chapter 4 Test Form 3 ..................................... 65
Chapter 4 Extended-Response Test ................ 67
Standardized Test Practice .............................. 68
Answers ........................................... A1–A34
Contents
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
iv
Teacher’s Guide to Using the
Chapter 4 Resource Masters
The Chapter 4 Resource Masters includes the core materials needed for Chapter 4. These
materials include worksheets, extensions, and assessment options. The answers for these
pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing, printing, and
editing at connectED.mcgraw-hill.com.
Chapter Resources
Student-Built Glossary (pages 1–2) These
masters are a student study tool that
presents up to twenty of the key vocabulary
terms from the chapter. Students are to
record definitions and/or examples for each
term. You may suggest that students
highlight or star the terms with which they
are not familiar. Give this to students before
beginning Lesson 4-1. Encourage them to
add these pages to their mathematics study
notebooks. Remind them to complete the
appropriate words as they study each lesson.
Anticipation Guide (pages 3–4) This
master, presented in both English and
Spanish, is a survey used before beginning
the chapter to pinpoint what students may
or may not know about the concepts in the
chapter. Students will revisit this survey
after they complete the chapter to see if
their perceptions have changed.
Lesson Resources
Study Guide and Intervention These
masters provide vocabulary, key concepts,
additional worked-out examples and Check
Your Progress exercises to use as a
reteaching activity. It can also be used in
conjunction with the Student Edition as an
instructional tool for students who have
been absent.
Skills Practice This master focuses more
on the computational nature of the lesson.
Use as an additional practice option or as
homework for second-day teaching of the
lesson.
Practice This master closely follows the
types of problems found in the Exercises
section of the Student Edition and includes
word problems. Use as an additional
practice option or as homework for second-
day teaching of the lesson.
Word Problem Practice This master
includes additional practice in solving word
problems that apply the concepts of the
lesson. Use as an additional practice or as
homework for second-day teaching of the
lesson.
Enrichment These activities may extend
the concepts of the lesson, offer an historical
or multicultural look at the concepts, or
widen students’ perspectives on the
mathematics they are learning. They are
written for use with all levels of students.
Graphing Calculator, TI-Nspire, or
Spreadsheet Activities
These activities present ways in which
technology can be used with the concepts in
some lessons of this chapter. Use as an
alternative approach to some concepts or as
an integral part of your lesson presentation.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
v
Assessment Options
The assessment masters in the Chapter 4
Resource Masters offer a wide range
of assessment tools for formative
(monitoring) assessment and summative
(final) assessment.
Student Recording Sheet This master
corresponds with the standardized test
practice at the end of the chapter.
Extended Response Rubric This master
provides information for teachers and stu-
dents on how to assess performance on open-
ended questions.
Quizzes Four free-response quizzes offer
assessment at appropriate intervals in
the chapter.
Mid-Chapter Test This 1-page test
provides an option to assess the first half of
the chapter. It parallels the timing of the
Mid-Chapter Quiz in the Student Edition
and includes both multiple-choice and free-
response questions.
Vocabulary Test This test is suitable for
all students. It includes a list of vocabulary
words and 11 questions to assess students’
knowledge of those words. This can also be
used in conjunction with one of the leveled
chapter tests.
Leveled Chapter Tests
• Form 1 contains multiple-choice ques-
tions and is intended for use with below
grade level students.
• Forms 2A and 2B contain multiple-
choice questions aimed at on grade level
students. These tests are similar in
format to offer comparable testing
situations.
• Forms 2C and 2D contain free-
response questions aimed at on grade
level students. These tests are similar in
format to offer comparable testing
situations.
• Form 3 is a free-response test for use
with above grade level students.
All of the above mentioned tests include a
free-response Bonus question.
Extended-Response Test Performance
assessment tasks are suitable for all
students. Sample answers and a scoring
rubric are included for evaluation.
Standardized Test Practice These three
pages are cumulative in nature. It includes
three parts: multiple-choice questions with
bubble-in answer format, griddable
questions with answer grids, and short-
answer free-response questions.
Answers
• The answers for the Anticipation Guide
and Lesson Resources are provided as
reduced pages.
• Full-size answer keys are provided for
the assessment masters.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter Resources
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
Chapter 4
1
Glencoe Algebra 1
Student-Built Glossary
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 4.
As you study the chapter, complete each term’s definition or description.
Remember to add the page number where you found the term. Add these pages to
your Algebra Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term
Found
on Page
Defi nition/Description/Example
best-fit line
bivariate data
correlation coefficient
kawr·uh·LAY·shun
inverse function
inverse relation
line of fit
linear extrapolation
ihk·STRA·puh·LAY·shun
linear interpolation
ihn·TUHR·puh·LAY·shun
(continued on the next page)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
Chapter 4
2
Glencoe Algebra 1
Student-Built Glossary (continued)
Vocabulary Term
Found
on Page
Defi nition/Description/Example
linear regression
median-fit line
parallel lines
perpendicular lines
PUHR·puhn·DIH·kyuh·luhr
residual
scatter plot
slope-intercept form
IHN·tuhr·SEHPT
Chapter Resources
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
Chapter 4
3
Glencoe Algebra 1
Anticipation Guide
Equations of Linear Functions
Before you begin Chapter 4
• Read each statement.
• Decide whether you Agree (A) or Disagree (D) with the statement.
• Write A or D in the first column OR if you are not sure whether you agree or
disagree, write NS (Not Sure).
After you complete Chapter 4
• Reread each statement and complete the last column by entering an A or a D.
• Did any of your opinions about the statements change from the first column?
• For those statements that you mark with a D, use a piece of paper to write an
example of why you disagree.
STEP 1
A, D, or NS
Statement
STEP 2
A or D
1. The slope of a line given by an equation in the form y = mx + b
can be determined by looking at the equation.
2. The y-intercept of y = 12x - 8 is 8.
3. If two points on a line are known, then an equation can be
written for that line.
4. An equation in the form y = mx + b is in point-slope form.
5. If a pair of lines are parallel, then they have the same slope.
6. Lines that intersect at right angles are called perpendicular
lines.
7. A scatter plot is said to have a negative correlation when the
points are random and show no relationship between x and y.
8. The closer the correlation coefficient is to zero, the more closely
a best-fit line models a set of data.
9. The equations of a regression line and a median-fit line are
very similar.
10. An inverse relation is obtained by exchanging the x-coordinates
with the y-coordinates of each ordered pair of the original
relation.
Step 1
Step 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Capítulo 4
4
Álgebra 1 de Glencoe
4
Ejercicios preparatorios
Ecuaciones de Funciones Lineales
Antes de comenzar el Capítulo 4
• Lee cada enunciado.
• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.
• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta,
escribe NS (No estoy seguro(a)).
Después de completar el Capítulo 4
• Vuelve a leer cada enunciado y completa la última columna con una A o una D.
• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?
• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los
enunciados que marcaste con una D.
PASO 1
A, D, o NS
Enunciado
PASO 2
A o D
1. La pendiente de una recta dada por una ecuación de la forma
y = mx + b se puede determinar mediante la observación de
la ecuación.
2. La intersección y de y = 12x - 8 es 8.
3. Si se conocen dos puntos sobre una recta, entonces se puede
escribir una ecuación para esa recta.
4. Una ecuación de la forma y = mx + b está en forma
punto-pendiente.
6. A las rectas que se intersecan en ángulos rectos se les
llama rectas perpendiculares.
7. Se dice que un diagrama de dispersión tiene correlación negativa
cuando los puntos son aleatorios y no muestran relación entre x y y.
8. Entre más cercano se encuentre de cero el coeficiente de
correlación, mejor modela un conjunto de datos la recta de
mejor ajuste.
9. La ecuación de una línea de regresión y una recta de mediano
ajuste son muy parecidas.
10. Una relación inversa es obtenida cambiando las x-coordenadas
con las y-coordenadas de cada par pedido de la relación original.
Paso 1
Paso 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-1
4-1
Chapter 4
5
Glencoe Algebra 1
Study Guide and Intervention
Graphing Equations in Slope-Intercept Form
Slope-Intercept Form
Slope-Intercept Form
y = mx + b, where m is the slope and b is the y-intercept
Write an equation in slope-intercept form for the line with a slope
of -4 and a y-intercept of 3.
y = mx + b
Slope-intercept form
y = -4x + 3
Replace m with -4 and b with 3.
Graph 3x - 4y = 8.
3x - 4y = 8
Original equation
-4y = -3x + 8
Subtract 3x from each side.
-4y
−
-4
= -3x + 8 −
-4
Divide each side by -4.
y = 3 −
4
x - 2
Simplify.
The y-intercept of y = 3 −
4
x - 2 is -2 and the slope is 3 −
4
. So graph the point (0, -2). From
this point, move up 3 units and right 4 units. Draw a line passing through both points.
Exercises
Write an equation of a line in slope-intercept form with the given slope and
y-intercept.
1. slope: 8, y-intercept -3
2. slope: -2, y-intercept -1
3. slope: -1, y-intercept -7
Write an equation in slope-intercept form for each graph shown.
4.
(0, –2)
(1, 0)
x
y
O
5.
(3, 0)
(0, 3)
x
y
O
6.
(4, –2)
(0, –5)
x
y
O
Graph each equation.
7. y = 2x + 1
8. y = -3x + 2
9. y = -x - 1
x
y
O
x
y
O
x
y
O
(0, –2)
(4, 1)
x
y
O
3x - 4y = 8
Example 1
Example 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4-1
Chapter 4
6
Glencoe Algebra 1
Study Guide and Intervention (continued)
Graphing Equations in Slope-Intercept Form
Modeling Real-World Data
MEDIA Since 1999, the number of music cassettes sold has
decreased by an average rate of 27 million per year. There were 124 million music
cassettes sold in 1999.
a. Write a linear equation to find the average number of music cassettes sold in
any year after 1999.
The rate of change is -27 million per year. In the first year, the number of music
cassettes sold was 124 million. Let N = the number of millions of music cassettes sold.
Let x = the number of years since 1999. An equation is N = -27x + 124.
b. Graph the equation.
The graph of N = -27x + 124 is a line that passes
through the point at (0, 124) and has a slope of -27.
c. Find the approximate number of music cassettes
sold in 2003.
N = -27x + 124
Original equation
N = -27(4) + 124
Replace x with 4.
N = 16
Simplify.
There were about 16 million music cassettes sold in 2003.
Exercises
1. MUSIC In 2001, full-length cassettes represented 3.4% of
total music sales. Between 2001 and 2006, the percent
decreased by about 0.5% per year.
a. Write an equation to find the percent P of recorded music
sold as full-length cassettes for any year x between
2001 and 2006.
b. Graph the equation on the grid at the right.
c. Find the percent of recorded music sold
as full-length cassettes in 2004.
2. POPULATION The population of the United States is
projected to be 300 million by the year 2010. Between
2010 and 2050, the population is expected to increase
by about 2.5 million per year.
a. Write an equation to find the population P in any year x
between 2010 and 2050.
b. Graph the equation on the grid at the right.
c. Find the population in 2050.
Full-length Cassette Sales
Percent of Total Music Sales
1.5%
2.0%
1.0%
2.5%
3.0%
3.5%
Years Since 2001
3
2
1
0
5
4
Projected United
States Population
Years Since 2010
Population (millions)
0
20
40
x
P
400
380
360
340
320
300
Music Cassettes Sold
Cassettes (millions)
50
75
25
0
100
125
Years Since 1999
3
2
1
5
7
4
6
Example
Lesson X-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-1
4-1
Chapter 4
7
Glencoe Algebra 1
Skills Practice
Graphing Equations in Slope-Intercept Form
Write an equation of a line in slope-intercept form with the given slope
and y-intercept.
1. slope: 5, y-intercept: -3
2. slope: -2, y-intercept: 7
3. slope: -6, y-intercept: -2
4. slope: 7, y-intercept: 1
5. slope: 3, y-intercept: 2
6. slope: -4, y-intercept: -9
7. slope: 1, y-intercept: -12
8. slope: 0, y-intercept: 8
Write an equation in slope-intercept form for each graph shown.
9.
(2, 1)
(0, –3)
x
y
O
10.
(0, 2)
(2, –4)
x
y
O
11.
(0, –1)
(2, –3)
x
y
O
Graph each equation.
12. y = x + 4
13. y = -2x - 1
14. x + y = -3
x
y
O
x
y
O
x
y
O
15. VIDEO RENTALS A video store charges $10 for a rental card
plus $2 per rental.
a. Write an equation in slope-intercept form for the total
cost c of buying a rental card and renting m
movies.
b. Graph the equation.
c. Find the cost of buying a rental card and renting 6 movies.
Video Store
Rental Costs
Total Cost ($)
10
0
12
14
16
18
20
c
Movies Rented
1
2
3
4
5 m
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4-1
Chapter 4
8
Glencoe Algebra 1
Practice
Graphing Equations in Slope-Intercept Form
Write an equation of a line in slope-intercept form with the given slope and
y-intercept.
1. slope: 1 −
4
, y-intercept: 3
2. slope: 3 −
2
, y-intercept: -4
3. slope: 1.5, y-intercept: -1
4. slope: -2.5, y-intercept: 3.5
Write an equation in slope-intercept form for each graph shown.
5.
(–5, 0)
(0, 2)
x
y
O
6.
(–2, 0)
(0, 3)
x
y
O
7.
(–3, 0)
(0, –2)
x
y
O
Graph each equation.
8. y = - 1 −
2
x + 2
9. 3y = 2x - 6
10. 6x + 3y = 6
x
y
O
x
y
O
x
y
O
11. WRITING Carla has already written 10 pages of a novel.
She plans to write 15 additional pages per month until she
is finished.
a. Write an equation to find the total number of pages P
written after any number of months m.
b. Graph the equation on the grid at the right.
c. Find the total number of pages written after 5 months.
Carla’s Novel
Months
Pages Written
2
0
4
6
1
3
5
m
P
100
80
60
40
20
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-1
4-1
Chapter 4
9
Glencoe Algebra 1
Word Problem Practice
Graphing Equations in Slope-Intercept Form
1. SAVINGS Wade’s grandmother gave him
$100 for his birthday. Wade wants to
save his money to buy a new MP3 player
that costs $275. Each month, he adds
$25 to his MP3 savings. Write an
equation in slope-intercept form for x, the
number of months that it will take Wade
to save $275.
2. CAR CARE Suppose regular gasoline
costs $2.76 per gallon. You can purchase
a car wash at the gas station for $3. The
graph of the equation for the cost of x
gallons of gasoline and a car wash is
shown below. Write the equation in slope-
intercept form for the line.
Gasoline (gal)
3
2
1
0
5
4
9
8
7
10
y
x
6
Cost of gas and car wash ($)
6
8
4
2
10
16
14
12
18
24
22
20
(4, 14.04)
(2, 8.52)
(0, 3)
3. ADULT EDUCATION Angie’s mother
wants to take some adult education
classes at the local high school. She has
to pay a one-time enrollment fee of $25
to join the adult education community,
and then $45 for each class she wants to
take. The equation y = 45x + 25
expresses the cost of taking x classes.
What are the slope and y-intercept of
the equation?
4. BUSINESS A construction crew needs to
rent a trench digger for up to a week. An
equipment rental company charges $40
per day plus a $20 non-refundable
insurance cost to rent a trench digger.
Write and graph an equation to find the
total cost to rent the trench digger for
d days.
Days
3
2
1
0
5
4
9
8
7
6
Price ($)
100
140
60
20
180
300
340
260
220
5. ENERGY From 2002 to 2005, U.S.
consumption of renewable energy
increased an average of 0.17 quadrillion
BTUs per year. About 6.07 quadrillion
BTUs of renewable power were produced
in the year 2002.
a. Write an equation in slope-intercept
form to find the amount of renewable
power P (quadrillion BTUs) produced
in year y between 2002 and 2005.
b. Approximately how much renewable
power was produced in 2005?
c. If the same trend continues from 2006
to 2010, how much renewable power
will be produced in the year 2010?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4-1
Chapter 4
10
Glencoe Algebra 1
Enrichment
Using Equations: Ideal Weight
You can find your ideal weight as follows.
A woman should weigh 100 pounds for the first 5 feet of height and
5 additional pounds for each inch over 5 feet (5 feet = 60 inches).
A man should weigh 106 pounds for the first 5 feet of height and
6 additional pounds for each inch over 5 feet. These formulas apply to
people with normal bone structures.
To determine your bone structure, wrap your thumb and index finger
around the wrist of your other hand. If the thumb and finger just touch,
you have normal bone structure. If they overlap, you are small-boned.
If they don’t overlap, you are large-boned. Small-boned people should decrease their
calculated ideal weight by 10%. Large-boned people should increase the value by 10%.
Calculate the ideal weights of these people.
1. woman, 5 ft 4 in., normal-boned
2. man, 5 ft 11 in., large-boned
3. man, 6 ft 5 in., small-boned
4. you, if you are at least 5 ft tall
For Exercises 5–9, use the following information.
Suppose a normal-boned man is x inches tall. If he is at least 5 feet
tall, then x - 60 represents the number of inches this man is over
5 feet tall. For each of these inches, his ideal weight is increased by
6 pounds. Thus, his proper weight y is given by the formula
y = 6(x - 60) + 106 or y = 6x - 254. If the man is large-boned, the
formula becomes y = 6x - 254 + 0.10(6x - 254).
5. Write the formula for the weight of a large-boned man
in slope-intercept form.
6. Derive the formula for the ideal weight y of a normal-boned
female with height x inches. Write the formula in
slope-intercept form.
7. Derive the formula in slope-intercept form for the ideal weight y
of a large-boned female with height x inches.
8. Derive the formula in slope-intercept form for the ideal weight y
of a small-boned male with height x inches.
9. Find the heights at which the ideal weights of normal-boned males
and large-boned females would be the same.
Lesson X-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-2
4-2
Chapter 4
11
Glencoe Algebra 1
Exercises
Write an equation of the line that passes through the given point and has the
given slope.
1.
(3, 5)
x
y
O
m = 2
2.
(0, 0)
x
y
O
m = –2
3.
(2, 4)
x
y
O
m = 12
4. (8, 2); slope - 3 −
4
5. (-1, -3); slope 5
6. (4, -5); slope - 1 −
2
7. (-5, 4); slope 0
8. (2, 2); slope 1 −
2
9. (1, -4); slope -6
10. (-3, 0), m = 2
11. (0, 4), m = -3
12. (0, 350), m = 1 −
5
Study Guide and Intervention
Writing Equations in Slope-Intercept Form
Write an Equation Given the Slope and a Point
Write an equation of
the line that passes through (-4, 2)
with a slope of 3.
The line has slope 3. To find the
y-intercept, replace m with 3 and (x, y)
with (-4, 2) in the slope-intercept form.
Then solve for b.
y = mx + b
Slope-intercept form
2 = 3(-4) + b
m = 3, y = 2, and x = -4
2 = -12 + b
Multiply.
14 = b
Add 12 to each side.
Therefore, the equation is y = 3x + 14.
Write an equation of the line
that passes through (-2, -1) with a
slope of 1 −
4
.
The line has slope 1 −
4
. Replace m with 1 −
4
and (x, y)
with (-2, -1) in the slope-intercept form.
y = mx + b
Slope-intercept form
-1 = 1 −
4
(-2) + b
m = 1
−
4
, y = -1, and x = -2
-1 = - 1 −
2
+ b
Multiply.
- 1 −
2
= b
Add 1
−
2
to each side.
Therefore, the equation is y = 1 −
4
x - 1 −
2
.
Example 1
Example 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4-2
Chapter 4
12
Glencoe Algebra 1
Study Guide and Intervention (continued)
Writing Equations in Slope-Intercept Form
Write an Equation Given Two Points
Write an equation of the line that passes through (1, 2) and (3, -2).
Find the slope m. To find the y-intercept, replace m with its computed value and (x, y) with
(1, 2) in the slope-intercept form. Then solve for b.
m =
y 2 - y 1 −
x 2 - x 1
Slope formula
m = -2 - 2 −
3 - 1
y2 = -2, y1 = 2, x2 = 3, x1 = 1
m = -2
Simplify.
y = mx + b
Slope-intercept form
2 = -2(1) + b
Replace m with -2, y with 2, and x with 1.
2 = -2 + b
Multiply.
4 = b
Add 2 to each side.
Therefore, the equation is y = -2x + 4.
Exercises
Write an equation of the line that passes through each pair of points.
1.
(1, 1)
(0, –3)
x
y
O
2.
(0, 4)
(4, 0)
x
y
O
3.
(0, 1)
(–3, 0)
x
y
O
4. (-1, 6), (7, -10)
5. (0, 2), (1, 7)
6. (6, -25), (-1, 3)
7. (-2, -1), (2, 11)
8. (10, -1), (4, 2)
9. (-14, -2), (7, 7)
10. (4, 0), (0, 2)
11. (-3, 0), (0, 5)
12. (0, 16), (-10, 0)
Example
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-2
Chapter 4
13
Glencoe Algebra 1
Write an equation of the line that passes through the given point with the
given slope.
1.
(–1, 4)
x
y
O
m = –3
2.
(4, 1)
x
y
O
m = 1
3.
(
-1, 2)
x
y
O
m = 2
4. (1, 9); slope 4
5. (4, 2); slope -2
6. (2, -2); slope 3
7. (3, 0); slope 5
8. (-3, -2); slope 2
9. (-5, 4); slope -4
Write an equation of the line that passes through each pair of points.
10.
(–2, 3)
(3, –2)
x
y
O
11.
(–1, –3)
(1, 1)
x
y
O
12.
(2, –1)
(0, 3)
x
y
O
13. (1, 3), (-3, -5)
14. (1, 4), (6, -1)
15. (1, -1), (3, 5)
16. (-2, 4), (0, 6)
17. (3, 3), (1, -3)
18. (-1, 6), (3, -2)
19. INVESTING The price of a share of stock in XYZ Corporation was $74 two weeks ago.
Seven weeks ago, the price was $59 a share.
a. Write a linear equation to find the price p of a share of XYZ Corporation stock
w weeks from now.
b. Estimate the price of a share of stock five weeks ago.
Skills Practice
Writing Equations in Slope-Intercept Form
4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
14
Glencoe Algebra 1
Practice
Writing Equations in Slope-Intercept Form
Write an equation of the line that passes through the given point and has the
given slope.
1.
(1, 2)
x
y
O
m = 3
2.
(–2, 2)
x
y
O
m = –2
3.
(–1, –3)
x
y
O
m = –1
4. (-5, 4); slope -3
5. (4, 3); slope 1 −
2
6. (1, -5); slope - 3 −
2
7. (3, 7); slope 2 −
7
8.
(
-2, 5 −
2
)
; slope - 1 − 2
9. (5, 0); slope 0
Write an equation of the line that passes through each pair of points.
10.
(4, –2)
(2, –4)
x
y
O
11.
(0, 5)
(4, 1)
x
y
O
12.
(–3, 1)
(–1, –3)
x
y
O
13. (0, -4), (5, -4)
14. (-4, -2), (4, 0)
15. (-2, -3), (4, 5)
16. (0, 1), (5, 3)
17. (-3, 0), (1, -6)
18. (1, 0), (5, -1)
19. DANCE LESSONS The cost for 7 dance lessons is $82. The cost for 11 lessons is $122.
Write a linear equation to find the total cost C for ℓ lessons. Then use the equation to
find the cost of 4 lessons.
20. WEATHER It is 76°F at the 6000-foot level of a mountain, and 49°F at the 12,000-foot
level of the mountain. Write a linear equation to find the temperature T at an elevation
x on the mountain, where x is in thousands of feet.
4-2
Lesson X-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-2
Chapter 4
15
Glencoe Algebra 1
Word Problem Practice
Writing Equations in Slope-Intercept Form
1. FUNDRAISING Yvonne and her friends
held a bake sale to benefit a shelter for
homeless people. The friends sold
22 cakes on the first day and 15 cakes on
the second day of the bake sale. They
collected $88 on the first day and $60 on
the second day. Let x represent the
number of cakes sold and y represent the
amount of money made. Find the slope of
the line that would pass through the
points given.
2. JOBS Mr. Kimball receives a $3000
annual salary increase on the
anniversary of his hiring if he receives
a satisfactory performance review.
His starting salary was $41,250. Write
an equation to show k, Mr. Kimball’s
salary after t years at this company
if his performance reviews are
always satisfactory.
3. CENSUS The population of Laredo,
Texas, was about 215,500 in 2007. It
was about 123,000 in 1990. If we assume
that the population growth is constant
and t represents the number of years
after 1990, write a linear equation to
find p, Laredo’s population for any year
after 1990.
4. WATER Mr. Williams pays $40 a month
for city water, no matter how many
gallons of water he uses in a given
month. Let x represent the number of
gallons of water used per month. Let y
represent the monthly cost of the city
water in dollars. What is the equation of
the line that represents this information?
What is the slope of the line?
5. SHOE SIZES The table shows how
women’s shoe sizes in the United
Kingdom compare to women’s shoe sizes
in the United States.
Women’s Shoe Sizes
U.K.
3
3.5
4
4.5
5
5.5
6
U.S. 5.5
6
6.5
7
7.5
8
8.5
Source: DanceSport UK
a. Write a linear equation to determine
any U.S. size y if you are given the
U.K. size x.
b. What are the slope and y-intercept of
the line?
c. Is the y-intercept a valid data point
for the given information?
4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
16
Glencoe Algebra 1
Tangent to a Curve
A tangent line is a line that intersects a curve at a point with the same rate of change, or
slope, as the rate of change of the curve at that point.
For quadratic functions, functions of the form y = ax2 + bx + c, equations of the tangent lines
can be found. This is based on the fact that the slope through any two points on the curve
is equal to the slope of the line tangent to the curve at the point whose x-value is halfway
between the x-values of the other two points.
Find an equation of the line tangent to the
curve y = x2 + 3x + 2 through the point (2, 12).
First find two points on the curve whose x-values are
equidistant from the x-value of (2, 12).
Step 1: Find two points on the curve. Use x = 1 and x = 3.
When x = 1, y = 12 + 3(1) + 2 or 6.
When x = 3, y = 32 + 3(3) + 2 or 20.
So, the two ordered pairs are (1, 6) and (3, 20).
Step 2: Find the slope of the line that passes through these two points.
m = 20 - 6 −
3 - 1
or 7
Step 3: Now use this slope and the point (2, 12) to find an equation of the tangent line.
y = mx + b
Slope-intercept form
12 = 7(2) + b
Replace x with 2, y with 12, and m with 7.
-2 = b
Solve for b.
So, an equation of the tangent line to y = x2 + 3x + 2 through the point (2, 12) is y = 7x – 2.
Exercises
Find an equation of the line tangent to each curve through the
given point.
1. y = x2 - 3x + 7, (2, 5)
2. y = 3x2 + 4x - 5, (-4, 27)
3. y = 5 - x2, (1, 4)
4. Find the slope of the line tangent to the curve at x = 0 for the general equation
y = ax2 + bx + c.
5. Find the slope of the line tangent to the curve y = ax2 + bx + c at x by finding the slope
of the line through the points (0, c) and (2x, 4ax2 + 2bx + c). Does this equation find the
same slope for x = 0 as you found in Exercise 4?
Enrichment
y
x
O
Example
4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-3
Chapter 4
17
Glencoe Algebra 1
Study Guide and Intervention
Writing Equations in Point-Slope Form
Point-Slope Form
Point-Slope Form
y - y1 = m(x - x1), where (x1, y1) is a given point on a nonvertical line
and m is the slope of the line
Write an equation in
point-slope form for the line that passes
through (6, 1) with a slope of -
5
− 2 .
y - y1 = m(x - x1)
Point-slope form
y - 1 = - 5 −
2
(x - 6) m = - 5 −
2
; (x1, y1) = (6, 1)
Therefore, the equation is y - 1 = - 5 −
2
(x - 6).
Write an equation in
point-slope form for a horizontal line
that passes through (4, -1).
y - y1 = m(x - x1)
Point-slope form
y - (-1) = 0(x - 4)
m = 0; (x1, y1) = (4, -1)
y + 1 = 0
Simplify.
Therefore, the equation is y + 1 = 0.
Exercises
Write an equation in point-slope form for the line that passes through each point
with the given slope.
1.
(4, 1)
x
y
O
m = 1
2.
(–3, 2)
x
y
O
m = 0
3.
(2, –3)
x
y
O
m = –2
4. (2, 1), m = 4
5. (-7, 2), m = 6
6. (8, 3), m = 1
7. (-6, 7), m = 0
8. (4, 9), m = 3 −
4
9. (-4, -5), m = - 1 −
2
10. Write an equation in point-slope form for a horizontal line that passes through
(4, -2).
11. Write an equation in point-slope form for a horizontal line that passes through
(-5, 6).
12. Write an equation in point-slope form for a horizontal line that passes through (5, 0).
Example 1
Example 2
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
18
Glencoe Algebra 1
Study Guide and Intervention (continued)
Writing Equations in Point-Slope Form
Forms of Linear Equations
Slope-Intercept
Form
y = mx + b
m = slope; b = y-intercept
Point-Slope
Form
y - y1 = m(x - x1)
m = slope; (x1, y1) is a given point
Standard
Form
Ax + By = C
A and B are not both zero. Usually A is nonnegative and A, B, and
C are integers whose greatest common factor is 1.
Write y + 5 = 2 −
3
(x - 6) in
standard form.
y + 5 = 2 −
3
(x - 6)
Original equation
3(y + 5) = 3 (
2 −
3
) (x - 6) Multiply each side by 3.
3y + 15 = 2(x - 6)
Distributive Property
3y + 15 = 2x - 12
Distributive Property
3y = 2x - 27
Subtract 15 from each side.
-2x + 3y = -27
Add -2x to each side.
2x - 3y = 27
Multiply each side by -1.
Therefore, the standard form of the equation
is 2x - 3y = 27.
Write y - 2 = - 1 −
4
(x - 8) in
slope-intercept form.
y - 2 = - 1 −
4
(x - 8)
Original equation
y - 2 = - 1 −
4
x + 2
Distributive Property
y = - 1 −
4
x + 4
Add 2 to each side.
Therefore, the slope-intercept form of the
equation is y = - 1 −
4
x + 4.
Exercises
Write each equation in standard form.
1. y + 2 = -3(x - 1)
2. y - 1 = - 1 −
3
(x - 6)
3. y + 2 = 2 −
3
(x - 9)
4. y + 3 = -(x - 5)
5. y - 4 = 5 −
3
(x + 3)
6. y + 4 = - 2 −
5
(x - 1)
Write each equation in slope-intercept form.
7. y + 4 = 4(x - 2)
8. y - 5 = 1 −
3
(x - 6)
9. y - 8 = - 1 −
4
(x + 8)
10. y - 6 = 3 (x -
1 −
3
)
11. y + 4 = -2(x + 5)
12. y + 5 −
3
= 1 −
2
(x - 2)
Example 1
Example 2
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-3
Chapter 4
19
Glencoe Algebra 1
Skills Practice
Writing Equations in Point-Slope Form
Write an equation in point-slope form for the line that passes through each point
with the given slope.
1.
(–1, –2)
x
y
O
m = 3
2.
(1, –2)
x
y
O
m = –1
3.
(2, –3)
x
y
O
m = 0
4. (3, 1), m = 0
5. (-4, 6), m = 8
6. (1, -3), m = -4
7. (4, -6), m = 1
8. (3, 3), m = 4 −
3
9. (-5, -1), m = - 5 −
4
Write each equation in standard form.
10. y + 1 = x + 2
11. y + 9 = -3(x - 2)
12. y - 7 = 4(x + 4)
13. y - 4 = -(x - 1)
14. y - 6 = 4(x + 3)
15. y + 5 = -5(x - 3)
16. y - 10 = -2(x - 3)
17. y - 2 = - 1 −
2
(x - 4)
18. y + 11 = 1 −
3
(x + 3)
Write each equation in slope-intercept form.
19. y - 4 = 3(x - 2)
20. y + 2 = -(x + 4)
21. y - 6 = -2(x + 2)
22. y + 1 = -5(x - 3)
23. y - 3 = 6(x - 1)
24. y - 8 = 3(x + 5)
25. y - 2 = 1 −
2
(x + 6)
26. y + 1 = - 1 −
3
(x + 9)
27. y - 1 −
2
= x + 1 −
2
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
20
Glencoe Algebra 1
Practice
Writing Equations in Point-Slope Form
Write an equation in point-slope form for the line that passes through each point
with the given slope.
1. (2, 2), m = -3
2. (1, -6), m = -1
3. (-3, -4), m = 0
4. (1, 3), m = - 3 −
4
5. (-8, 5), m = - 2 −
5
6. (3, -3), m = 1 −
3
Write each equation in standard form.
7. y - 11 = 3(x - 2)
8. y - 10 = -(x - 2)
9. y + 7 = 2(x + 5)
10. y - 5 = 3 −
2
(x + 4)
11. y + 2 = - 3 −
4
(x + 1)
12. y - 6 = 4 −
3
(x - 3)
13. y + 4 = 1.5(x + 2)
14. y - 3 = -2.4(x - 5)
15. y - 4 = 2.5(x + 3)
Write each equation in slope-intercept form.
16. y + 2 = 4(x + 2)
17. y + 1 = -7(x + 1)
18. y - 3 = -5(x + 12)
19. y - 5 = 3 −
2
(x + 4)
20. y - 1 −
4
= - 3 (x +
1 −
4
)
21. y - 2 −
3
= -2 (x -
1 −
4
)
22. CONSTRUCTION A construction company charges $15 per hour for debris removal,
plus a one-time fee for the use of a trash dumpster. The total fee for 9 hours of service
is $195.
a. Write the point-slope form of an equation to find the total fee y for any number of
hours x.
b. Write the equation in slope-intercept form.
c. What is the fee for the use of a trash dumpster?
23. MOVING There is a daily fee for renting a moving truck, plus a charge of $0.50 per mile
driven. It costs $64 to rent the truck on a day when it is driven 48 miles.
a. Write the point-slope form of an equation to find the total charge y for a one-day
rental with x miles driven.
b. Write the equation in slope-intercept form.
c. What is the daily fee?
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-3
Chapter 4
21
Glencoe Algebra 1
1. BICYCLING Harvey rides his bike at an
average speed of 12 miles per hour. In
other words, he rides 12 miles in 1 hour,
24 miles in 2 hours, and so on. Let h be
the number of hours he rides and d be
distance traveled. Write an equation for
the relationship between distance and
time in point-slope form.
2. GEOMETRY The perimeter of a square
varies directly with its side length. The
point-slope form of the equation for this
function is y - 4 = 4(x - 1). Write the
equation in standard form.
3. NATURE The frequency of a male
cricket’s chirp is related to the outdoor
temperature. The relationship is
expressed by the equation T = n + 40,
where T is the temperature in degrees
Fahrenheit and n is the number of chirps
the cricket makes in 14 seconds. Use
the information from the graph below to
write an equation for the line in point-
slope form .
Number of Chirps
15
10
5
0
25
20
y
x
30 35
Temperature (°F)
30
40
20
10
50
70
60
4. CANOEING Geoff paddles his canoe at
an average speed of 3.5 miles per hour.
After 5 hours of canoeing, Geoff has
traveled 18 miles. Write an equation in
point-slope form to find the total distance
y for any number of hours x.
5. AVIATION A jet plane takes off and
consistently climbs 20 feet for every
40 feet it moves horizontally. The graph
shows the trajectory of the jet.
Horizontal Distance (ft)
500
0
1000
1500 2000
2500
Height (ft)
600
800
400
200
1000
1400
1200
a. Write an equation in point-slope form
for the line representing the jet’s
trajectory.
b. Write the equation from part a in
slope -intercept form.
c. Write the equation in standard form.
Word Problem Practice
Writing Equations in Point-Slope Form
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
22
Glencoe Algebra 1
Enrichment
x
y
O
x
y
O
Collinearity
You have learned how to find the slope between two points on a line. Does
it matter which two points you use? How does your choice of points affect
the slope-intercept form of the equation of the line?
1. Choose three different pairs of points from the graph at the
right. Write the slope-intercept form of the line using each pair.
2. How are the equations related?
3. What conclusion can you draw from your answers to Exercises 1 and 2?
When points are contained in the same line, they are said to be collinear.
Even though points may look like they form a line when connected, it does
not mean that they actually do. By checking pairs of points on a graph
you can determine whether the graph represents a linear relationship.
4. Choose several pairs of points from the graph at the right
and write the slope-intercept form of the line containing
each pair.
5. What conclusion can you draw from your equations in
Exercise 4? Is this a line?
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-3
Chapter 4
23
Glencoe Algebra 1
Graphing Calculator Activity
Writing Linear Equations
Lists can be used with the linear regression function to write and verify
linear equations given two points on a line, or the slope of a line and a point
through which it passes. The linear regression function, LinReg (ax + b), is
found under the STAT CALC menu.
Write the slope-intercept form of an equation of the
line that passes through (3, -2) and (6, 4).
Enter the x-coordinates of the points into L1 and the y-coordinates
into L2. Use the linear regression function to write the equation of
the line.
Keystrokes: STAT ENTER 3 ENTER 6 ENTER
(–) 2 ENTER 4
ENTER STAT
4 2nd
[L1]
, 2nd
[L2] ENTER .
The equation is y = 2x - 8.
If you have already written the equation of a line, you can use the given information to
verify your equation.
Exercises
Write the slope-intercept form and the standard form of an equation of the line
that satisfies each condition.
1. passes through (0, 7) and (
1 −
7
, -5)
2. passes through (-5, 1), (10, 10), and (-10, -2)
3. passes through (6, -4), m = 2 −
3
4. passes through (3, 5), m = -4
5. x-intercept: 1, y-intercept: - 1 −
2
6. passes through (-18, 11), y-intercept: 3
Verify that the equation of the line passing through
(2, -3) with slope -
3
− 4 can be written as 3x + 4y = -6.
Use the given point and slope to determine a second point through
which the line passes. Enter the x-coordinates of the points into L1
and the y-coordinates into L2. Use LinReg (ax + b) to determine
the slope-intercept form of the equation.
The slope-intercept form of the equation is y = -0.75x - 1.5 or y = - 3 −
4
x - 3 −
2
.
This can be rewritten in standard form as 3x + 4y = -6.
Example 1
Example 2
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
24
Glencoe Algebra 1
Study Guide and Intervention
Parallel and Perpendicular Lines
Parallel Lines Two nonvertical lines are parallel if they have the same slope. All
vertical lines are parallel.
Write an equation in slope-intercept form for the line that passes
through (-1, 6) and is parallel to the graph of y = 2x + 12.
A line parallel to y = 2x + 12 has the same slope, 2. Replace m with 2 and (x1, y1) with
(-1, 6) in the point-slope form.
y - y1 = m(x - x1)
Point-slope form
y - 6 = 2(x - (-1)) m = 2; (x1, y1) = (-1, 6)
y - 6 = 2(x + 1)
Simplify.
y - 6 = 2x + 2
Distributive Property
y = 2x + 8
Slope-intercept form
Therefore, the equation is y = 2x + 8.
Exercises
Write an equation in slope-intercept form for the line that passes through the
given point and is parallel to the graph of each equation.
1.
2.
3.
4. (-2, 2), y = 4x - 2
5. (6, 4), y = 1 −
3
x + 1
6. (4, -2), y = -2x + 3
7. (-2, 4), y = -3x + 10
8. (-1, 6), 3x + y = 12
9. (4, -6), x + 2y = 5
10. Find an equation of the line that has a y-intercept of 2 that is parallel to the graph of
the line 4x + 2y = 8.
11. Find an equation of the line that has a y-intercept of -1 that is parallel to the graph of
the line x - 3y = 6.
12. Find an equation of the line that has a y-intercept of -4 that is parallel to the graph of
the line y = 6.
(–3, 3)
x
y
O
4x - 3y = –12
(
-8, 7)
x
y
O
y = - x - 4
1
2
2
2
(5, 1)
x
y
O
y = x - 8
Example
4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-4
Chapter 4
25
Glencoe Algebra 1
Study Guide and Intervention (continued)
Parallel and Perpendicular Lines
Perpendicular Lines Two nonvertical lines are perpendicular if their slopes are
negative reciprocals of each other. Vertical and horizontal lines are perpendicular.
Write an equation in slope-intercept form for the line that passes
through (-4, 2) and is perpendicular to the graph of 2x - 3y = 9.
Find the slope of 2x - 3y = 9.
2x - 3y = 9
Original equation
-3y = -2x + 9 Subtract 2x from each side.
y = 2 −
3
x - 3
Divide each side by -3.
The slope of y = 2 −
3
x - 3 is 2 −
3
. So, the slope of the line passing through (-4, 2) that is
perpendicular to this line is the negative reciprocal of 2 −
3
, or - 3 −
2
.
Use the point-slope form to find the equation.
y - y1 = m(x - x1)
Point-slope form
y - 2 = - 3 −
2
(x - (-4))
m = - 3 −
2
; (x1, y1) = (-4, 2)
y - 2 = - 3 −
2
(x + 4)
Simplify.
y - 2 = - 3 −
2
x - 6
Distributive Property
y = - 3 −
2
x - 4
Slope-intercept form
Exercises
1. ARCHITECTURE On the architect’s plans for a new high school, a wall represented
by
−−−
MN has endpoints M(-3, -1) and N(2, 1). A wall represented by
−−−
PQ has endpoints
P(4, -4) and Q(-2, 11). Are the walls perpendicular? Explain.
Determine whether the graphs of the following equations are parallel or
perpendicular.
2. 2x + y = -7, x - 2y = -4, 4x - y = 5
3. y = 3x, 6x - 2y = 7, 3y = 9x - 1
Write an equation in slope-intercept form for the line that passes through the
given point and is perpendicular to the graph of each equation.
4. (4, 2), y = 1 −
2
x + 1
5. (2, -3), y = - 2 −
3
x + 4
6. (6, 4), y = 7x + 1
7. (-8, -7), y = -x - 8
8. (6, -2), y = -3x - 6
9. (-5, -1), y = 5 −
2
x - 3
Example
4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
26
Glencoe Algebra 1
Skills Practice
Parallel and Perpendicular Lines
Write an equation in slope-intercept form for the line that passes through the
given point and is parallel to the graph of the given equation.
1.
2.
3.
4. (3, 2), y = 3x + 4
5. (-1, -2), y = -3x + 5
6. (-1, 1), y = x - 4
7. (1, -3), y = -4x - 1
8. (-4, 2), y = x + 3
9. (-4, 3), y = 1 −
2
x - 6
10. RADAR On a radar screen, a plane located at A(-2, 4) is flying toward B(4, 3).
Another plane, located at C(-3, 1), is flying toward D(3, 0). Are the planes’ paths
perpendicular? Explain.
Determine whether the graphs of the following equations are parallel or
perpendicular. Explain.
11. y = 2 −
3
x + 3, y = 3 −
2
x, 2x - 3y = 8
12. y = 4x, x + 4 y = 12, 4x + y = 1
Write an equation in slope-intercept form for the line that passes through the
given point and is perpendicular to the graph of the given equation.
13. (-3, -2), y = x + 2
14. (4, -1), y = 2x - 4
15. (-1, -6), x + 3y = 6
16. (-4, 5), y = -4x - 1
17. (-2, 3), y = 1 −
4
x - 4
18. (0, 0), y = 1 −
2
x - 1
(–2, 2)
x
y
O
y = 12 x + 1
(1, –1)
x
y
O
y = –x + 3
(–2, –3)
x
y
O
y = 2x - 1
4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-4
Chapter 4
27
Glencoe Algebra 1
Practice
Parallel and Perpendicular Lines
Write an equation in slope-intercept form for the line that passes through the
given point and is parallel to the graph of the given equation.
1. (3, 2), y = x + 5
2. (-2, 5), y = -4x + 2
3. (4, -6), y = - 3 −
4
x + 1
4. (5, 4), y = 2 −
5
x - 2
5. (12, 3), y = 4 −
3
x + 5
6. (3, 1), 2x + y = 5
7. (-3, 4), 3y = 2x - 3
8. (-1, -2), 3x - y = 5
9. (-8, 2), 5x - 4y = 1
10. (-1, -4), 9x + 3y = 8
11. (-5, 6), 4x + 3y = 1
12. (3, 1), 2x + 5y = 7
Write an equation in slope-intercept form for the line that passes through the
given point and is perpendicular to the graph of the given equation.
13. (-2, -2), y = - 1 −
3
x + 9
14. (-6, 5), x - y = 5
15. (-4, -3), 4x + y = 7
16. (0, 1), x + 5y = 15
17. (2, 4), x - 6y = 2
18. (-1, -7), 3x + 12y = -6
19. (-4, 1), 4x + 7y = 6
20. (10, 5), 5x + 4y = 8
21. (4, -5), 2x - 5y = -10
22. (1, 1), 3x + 2y = -7
23. (-6, -5), 4x + 3y = -6
24. (-3, 5), 5x - 6y = 9
25. GEOMETRY Quadrilateral ABCD has diagonals −−
AC and
−−−
BD .
Determine whether
−−
AC is perpendicular to
−−−
BD . Explain.
26. GEOMETRY Triangle ABC has vertices A(0, 4), B(1, 2), and C(4, 6). Determine whether
triangle ABC is a right triangle. Explain.
x
y
O
A
D
C
B
4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
28
Glencoe Algebra 1
1. BUSINESS Brady’s Books is a retail
store. The store’s average daily profits y
are given by the equation y = 2x + 3
where x is the number of hours available
for customer purchases. Brady’s adds an
online shopping option. Write an
equation in slope-intercept form to show
a new profit line with the same profit
rate containing the point (0, 12).
2. ARCHITECTURE The front view of a
house is drawn on graph paper. The left
side of the roof of the house is
represented by the equation y = x. The
rooflines intersect at a right angle and
the peak of the roof is represented by the
point (5, 5). Write the equation in slope-
intercept form for the line that creates
the right side of the roof.
3. ARCHAEOLOGY An archaeologist is
comparing the location of a jeweled box
she just found to the location of a brick
wall. The wall can be represented by the
equation y = - 5 −
3
x + 13. The box is
located at the point (10, 9). Write an
equation representing a line that is
perpendicular to the wall and that passes
through the location of the box.
4. GEOMETRY A parallelogram is created
by the intersections of the lines x = 2,
x = 6, y = 1 −
2
x + 2, and another line. Find
the equation of the fourth line needed to
complete the parallelogram. The line
should pass through (2, 0). (Hint: Sketch
a graph to help you see the lines.)
5. INTERIOR DESIGN Pamela is planning
to install an island in her kitchen. She
draws the shape she likes by connecting
the vertices of the square tiles on her
kitchen floor. She records the location of
each corner in the table.
a. How many pairs of parallel sides are
there in the shape ABCD she
designed? Explain.
b. How many pairs of perpendicular
sides are there in the shape she
designed? Explain.
c. What is the shape of her new island?
Word Problem Practice
Parallel and Perpendicular Lines
y
x
O
(5, 5)
Corner
Distance
from West
Wall (tiles)
Distance
from South
Wall (tiles)
A
5
4
B
3
8
C
7
10
D
11
7
4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-4
Chapter 4
29
Glencoe Algebra 1
Enrichment
Pencils of Lines
All of the lines that pass through
a single point in the same plane
are called a pencil of lines.
All lines with the same slope,
but different intercepts, are also
called a “pencil,” a pencil of
parallel lines.
Graph some of the lines in each pencil.
1. A pencil of lines through the
2. A pencil of lines described by
point (1, 3)
y - 4 = m(x - 2), where m is any
real number
3. A pencil of lines parallel to the line
4. A pencil of lines described by
x - 2y = 7
y = mx + 3m - 2 , where m is any
real number
x
y
O
x
y
O
x
y
O
x
y
O
4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
30
Glencoe Algebra 1
Study Guide and Intervention
Scatter Plots and Lines of Fit
Investigate Relationships Using Scatter Plots A scatter plot is a graph in
which two sets of data are plotted as ordered pairs in a coordinate plane. If y increases as x
increases, there is a positive correlation between x and y. If y decreases as x increases,
there is a negative correlation between x and y. If x and y are not related, there is no
correlation.
EARNINGS The graph at the right
shows the amount of money Carmen earned each
week and the amount she deposited in her savings
account that same week. Determine whether the
graph shows a positive correlation, a negative
correlation, or no correlation. If there is a
positive or negative correlation, describe its
meaning in the situation.
The graph shows a positive correlation. The more
Carmen earns, the more she saves.
Exercises
Determine whether each graph shows a positive correlation, a negative
correlation, or no correlation. If there is a positive or negative correlation,
describe its meaning in the situation.
1.
2.
3.
4.
Average Weekly Work Hours in U.S.
Hours
34.0
34.2
33.8
33.6
34.4
34.6
Years Since 1995
3
2
1
0
5
4
7
6
9
8
Source: The World Almanac
Average Jogging Speed
Minutes
Miles per Hour
0
10
20
5
15
25
10
5
Carmen’s Earnings and Savings
Dollars Earned
Dollars Saved
0
40
80
120
35
30
25
20
15
10
5
Example
Average U.S. Hourly
Earnings
Hourly Earnings ($)
15
0
16
17
18
19
Years Since 2003
Source: U.S. Dept. of Labor
1
2
3
4
5
U.S. Imports from Mexico
Imports ($ billions)
130
0
160
190
220
Years Since 2003
Source: U.S. Census Bureau
1
2
3
4
5
4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-5
Chapter 4
31
Glencoe Algebra 1
Use Lines of Fit
The table shows the number of students per computer in Easton
High School for certain school years from 1996 to 2008.
Year
1996
1998
2000
2002
2004
2006
2008
Students per Computer
22
18
14
10
6.1
5.4
4.9
a. Draw a scatter plot and determine
what relationship exists, if any.
Since y decreases as x increases, the
correlation is negative.
b. Draw a line of fit for the scatter plot.
Draw a line that passes close to most of the points.
A line of fit is shown.
c. Write the slope-intercept form of an equation
for the line of fit.
The line of fit shown passes through
(1999, 16) and (2005, 5.7). Find the slope.
m = 5.7 - 16
−
2005 - 1999
m = -1.7
Find b in y = -1.7x + b.
16 = -1.7 · 1993 + b
3404 = b
Therefore, an equation of a line of fit is y = -1.7x + 3404.
Exercises
Refer to the table for Exercises 1–3.
1. Draw a scatter plot.
2. Draw a line of fit for the data.
3. Write the slope-intercept
form of an equation for the
line of fit.
Movie Admission Prices
Admission ($)
5.4
5.6
5.2
5
5.8
6
6.2
Years Since 1999
3
2
1
0
5
4
Source: U.S. Census Bureau
Study Guide and Intervention (continued)
Scatter Plots and Lines of Fit
Students per Computer
in Easton High School
Students per Computer
8
12
16
4
0
20
24
Year
1996 1998 2000 2002 2004 2006 2008
Example
Years
Since 1999
Admission
(dollars)
0
$5.08
1
$5.39
2
$5.66
3
$5.81
4
$6.03
4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
32
Glencoe Algebra 1
Skills Practice
Scatter Plots and Lines of Fit
Determine whether each graph shows a positive correlation, a negative
correlation, or no correlation. If there is a positive or negative correlation,
describe its meaning in the situation.
1.
2.
3.
4.
5. BASEBALL The scatter plot shows the average price of a major-league baseball ticket
from 1997 to 2006.
a. Determine what relationship, if any, exists in the
data. Explain.
b. Use the points (1998, 13.60) and (2003, 19.00)
to write the slope-intercept form of an equation for
the line of fit shown in the scatter plot.
c. Predict the price of a ticket in 2009.
Weight-Lifting
Weight (pounds)
Repetitions
0
40
80
20
60
100 120 140
14
12
10
8
6
4
2
Library Fines
Books Borrowed
Fines (dollars)
0
2
4
5
6
7
8
9
1
3
10
7
6
5
4
3
2
1
Calories Burned
During Exercise
Time (minutes)
Calories
0
20
40
10
30
50 60
600
500
400
300
200
100
Baseball Ticket Prices
Average Price ($)
14
16
12
0
18
20
22
24
Year
’99
’98
’97
’01
’03
’00
Source: Team Marketing Report, Chicago
’02
’04 ’05 ’06
Car Dealership Revenue
Revenue
(hundreds of thousands)
4
6
2
0
8
10
12
14
Year
’99
’01
’03
’00
’02
’04 ’05 ’06 ’07 ’08
4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-5
Chapter 4
33
Glencoe Algebra 1
Practice
Scatter Plots and Lines of Fit
Determine whether each graph shows a positive correlation, a negative
correlation, or no correlation. If there is a positive or negative correlation,
describe its meaning in the situation.
1.
2.
3. DISEASE The table shows the number of cases of
Foodborne Botulism in the United States for the
years 2001 to 2005.
a. Draw a scatter plot and determine what
relationship, if any, exists in the data.
b. Draw a line of fit for the scatter plot.
c. Write the slope-intercept form of an equation for the
line of fit.
4. ZOOS The table shows the average and maximum
longevity of various animals in captivity.
a. Draw a scatter plot and determine what
relationship, if any, exists in the data.
b. Draw a line of fit for the scatter plot.
c. Write the slope-intercept form of an equation for the
line of fit.
d. Predict the maximum longevity for an animal with
an average longevity of 33 years.
State Elevations
Mean Elevation (feet)
Highest Point
(thousands of feet)
1000
0
2000
3000
16
12
8
4
Source: U.S. Geological Survey
Temperature versus Rainfall
Average Annual Rainfall (inches)
Average
Temperature (ºF)
10 15 20 25 30 35 40 45
64
60
56
52
0
Source: National Oceanic and Atmospheric
Administration
U.S. Foodborne
Botulism Cases
Cases
20
30
10
0
40
50
Year
2001
2002
2003
2004
2005
Animal Longevity (Years)
Average
Maximum
5
0
10 15 20 25 30 35 40 45
80
70
60
50
40
30
20
10
Source: Centers for Disease Control
U.S. Foodborne Botulism Cases
Year
2001 2002 2003 2004 2005
Cases
39
28
20
16
18
Source: Walker’s Mammals of the World
Longevity (years)
Avg. 12 25 15
8 35 40 41 20
Max. 47 50 40 20 70 77 61 54
4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
34
Glencoe Algebra 1
1. MUSIC The scatter plot shows the
number of CDs in millions that were sold
from 1999 to 2005. If the trend
continued, about how many CDs were
sold in 2006?
2. FAMILY The table shows the predicted
annual cost for a middle income family to
raise a child from birth until adulthood.
Draw a scatter plot and describe what
relationship exists within the data.
3. HOUSING The median price of an
existing home was $160,000 in 2000 and
$240,000 in 2007. If x represents the
number of years since 2000, use these
data points to determine a line of best fit
for the trends in the price of existing
homes. Write the equation in slope-
intercept form.
4. BASEBALL The table shows the average
length in minutes of professional
baseball games in selected years.
Source: Elias Sports Bureau
a. Draw a scatter plot and determine
what relationship, if any, exists in the
data.
b. Explain what the scatter plot shows.
c. Draw a line of fit for the scatter plot.
Time (min)
166
0
168
170
172
174
176
178
180
Year
’90 ’92 ’94 ’96 ’98 ’00 ’02
Age (years)
3
0
6
12
15
y
x
9
Annual Cost ($1000)
11
12
10
9
13
16
15
14
17
Source: The World Almanac
Source: RIAA
Year
‘01
‘00
‘99
0
‘03
‘02
‘05
y
x
‘04
CDs (millions)
750
800
700
650
850
950
900
Word Problem Practice
Scatter Plots and Lines of Fit
Cost of Raising a Child Born in 2003
Child’s
Age
3
6
9
12
15
Annual
Cost ($)
10,700 11,700 12,600 15,000 16,700
Average Length of
Major League Baseball Games
Year
‘92
‘94
‘96
‘98
‘00
‘02
‘04
Time (min) 170 174 171 168 178 172 167
4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-5
Chapter 4
35
Glencoe Algebra 1
Enrichment
Latitude and Temperature
The latitude of a place on Earth
is the measure of its distance from
the equator. What do you think is
the relationship between a city’s
latitude and its mean January
temperature? At the right is a
table containing the latitudes and
January mean temperatures for
fifteen U.S. cities.
Sources: National Weather Service
1. Use the information in the table to create
a scatter plot and draw a line of best fit
for the data.
2. Write an equation for the line of fit. Make
a conjecture about the relationship
between a city’s latitude and its mean
January temperature.
3. Use your equation to predict the January
mean temperature of Juneau, Alaska, which has latitude 58:23 N.
4. What would you expect to be the latitude of a city with a January mean temperature
of 15°F?
5. Was your conjecture about the relationship between latitude and temperature correct?
6. Research the latitudes and temperatures for cities in the southern hemisphere. Does
your conjecture hold for these cities as well?
Latitude (ºN)
Temperature (
º
F)
70
60
50
40
30
20
10
0
-10
T
L
20
40
60
10
30
50
U.S. City
Latitude January Mean Temperature
Albany, New York
42:40 N
20.7°F
Albuquerque, New Mexico
35:07 N
34.3°F
Anchorage, Alaska
61:11 N
14.9°F
Birmingham, Alabama
33:32 N
41.7°F
Charleston, South Carolina 32:47 N
47.1°F
Chicago, Illinois
41:50 N
21.0°F
Columbus, Ohio
39:59 N
26.3°F
Duluth, Minnesota
46:47 N
7.0°F
Fairbanks, Alaska
64:50 N
-10.1°F
Galveston, Texas
29:14 N
52.9°F
Honolulu, Hawaii
21:19 N
72.9°F
Las Vegas, Nevada
36:12 N
45.1°F
Miami, Florida
25:47 N
67.3°F
Richmond, Virginia
37:32 N
35.8°F
Tucson, Arizona
32:12 N
51.3°F
4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
36
Glencoe Algebra 1
Exercises
The table shows the number of millions of dollars of direct
political contributions received by Democrats and Republicans
in selected years.
1. Use a spreadsheet to draw a scatter plot and a trendline for the
data. Let x represent the number of years since 1990 and let
y represent direct political contributions in millions of dollars.
2. Predict the amount of direct political contributions for the 2010 election.
Spreadsheet Activity
Scatter Plots
The table below shows the number of metric tons of gold produced
in mines in the United States in selected years.
Use a spreadsheet to draw a scatter plot and a trendline for the data.
Let x represent the number of years since 2000 and let y represent the
number of metric tons of gold. Then predict the number of ounces of
gold produced in 2013.
Step 1 Use Column A for the years since 2000 and Column B for the number of metric
tons of gold. To create a graph from the data, select the data in Columns A and B
and choose Chart from the Insert menu. Select an XY (Scatter) chart to show the
data points.
Step 2 Add a trendline to the graph by choosing the Chart menu. Add a linear
trendline. Use the options menu to have the trendline forecast 5 years forward.
Using this trendline, it appears that the gold production for 2013 will be
approximately 150 metric tons.
A spreadsheet program can create scatter plots of data that you enter. You can also
have the spreadsheet graph a line of fit, called a trendline, automatically.
Example
Source: Open Secrets
Year
Contributions
1990
281
1994
337
1998
445
2002
717
2006
1085
Source: U.S. Geological Survey
Year
2000 2001
2002
2003
2004 2005
2006
2007
2008
2009
Gold
353
335
298
277
247
256
252
238
233
210
4-5
A
1
0
1
2
3
4
5
6
7
8
9
353
335
298
277
247
256
252
238
233
210
3
4
5
6
7
8
9
10
11
12
13
14
2
B
C
D
E
F
G
H
15
Spreadsheet sample
Sheet 1
Sheet 2
Sheet 3
U.S. Gold Mine Production
Gold (metric tons)
100
150
50
0
200
250
300
350
400
Years since 2000
5
10
15
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-6
Chapter 4
37
Glencoe Algebra 1
Equations of Best-Fit Lines Many graphing calculators utilize an algorithm called
linear regression to find a precise line of fit called the best-fit line. The calculator
computes the data, writes an equation, and gives you the correlation coefficent, a
measure of how closely the equation models the data.
GAS PRICES The table shows the price of a gallon of regular
gasoline at a station in Los Angeles, California on January 1 of various years.
Year
2005
2006
2007
2008
2009
2010
Average Price
$1.47
$1.82
$2.15
$2.49
$2.83
$3.04
Source: U.S. Department of Energy
a. Use a graphing calculator to write an equation for the
best-fit line for that data. Enter the data by pressing STAT
and selecting the Edit option. Let the year 2005 be represented
by 0. Enter the years since 2005 into List 1 (L1). Enter the
average price into List 2 (L2).
Then, perform the linear regression by pressing STAT and
selecting the CALC option. Scroll down to LinReg (ax+b) and
press ENTER . The best-fit equation for the regression is shown
to be y = 0.321x + 1.499.
b. Name the correlation coefficient. The correlation coefficient
is the value shown for r on the calculator screen. The correlation
coefficient is about 0.998.
Exercises
Write an equation of the regression line for the data in each table below. Then
find the correlation coefficient.
1. OLYMPICS Below is a table showing the number of gold medals won by the United
States at the Winter Olympics during various years.
Year
1992
1994
1998
2002
2006
2010
Gold Medals
5
6
6
10
9
9
Source: International Olympic Committee
2. INTEREST RATES Below is a table showing the U.S. Federal Reserve’s prime interest
rate on January 1 of various years.
Year
2006
2007
2008
2009
2010
Prime Rate (percent)
7.25
8.25
7.25
3.25
3.25
Source: Federal Reserve Board
Study Guide and Intervention
Regression and Median-Fit Lines
Example
4-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
38
Glencoe Algebra 1
Equations of Median-Fit Lines A graphing calculator can also find another type of
best-fit line called the median-fit line, which is found using the medians of the coordinates
of the data points.
ELECTIONS The table shows the total number of people in millions
who voted in the U.S. Presidential election in the given years.
Year
1980
1984
1988
1992
1996
2004
2008
Voters
86.5
92.7
91.6
104.4
96.3
122.3
131.3
Source: George Mason University
a. Find an equation for the median-fit line. Enter the data by
pressing STAT and selecting the Edit option. Let the year 1980
be represented by 0. Enter the years since 1980 into List 1 (L1).
Enter the number of voters into List 2 (L2).
Then, press STAT and select the CALC option. Scroll down to
Med-Med option and press ENTER . The value of a is the slope,
and the value of b is the y-intercept.
The equation for the median-fit line is y = 1.55x + 83.57.
b. Estimate the number of people who voted in the 2000 U.S.
Presidential election. Graph the best-fit line. Then use the
TRACE feature and the arrow keys until you find a point
where x = 20.
When x = 20, y ≈ 115. Therefore, about 115 million people voted in the 2000 U.S.
Presidential election.
Exercises
Write an equation of the regression line for the data in each table below. Then
find the correlation coefficient.
1. POPULATION GROWTH Below is a table showing the estimated population of Arizona
in millions on July 1st of various years.
Year
2001
2002
2003
2004
2005
2006
Population
5.30
5.44
5.58
5.74
5.94
6.17
Source: U.S. Census Bureau
a. Find an equation for the median-fit line.
b. Predict the population of Arizona in 2009.
2. ENROLLMENT Below is a table showing the number of students enrolled at Happy
Days Preschool in the given years.
Year
2002
2004
2006
2008
2010
Students
130
168
184
201
234
a. Find an equation for the median-fit line.
b. Estimate how many students were enrolled in 2007.
Study Guide and Intervention (continued)
Regression and Median-Fit Lines
Example
4-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-6
Chapter 4
39
Glencoe Algebra 1
Write an equation of the regression line for the data in each table below. Then
find the correlation coefficient.
1. SOCCER The table shows the number of goals a soccer team scored each season
since 2005.
Year
2005
2006
2007
2008
2009
2010
Goals Scored
42
48
46
50
52
48
2. PHYSICAL FITNESS The table shows the percentage of seventh grade students
in public school who met all six of California’s physical fitness standards each year
since 2002.
Year
2002
2003
2004
2005
2006
Percentage
24.0%
36.4%
38.0%
40.8%
37.5%
Source: California Department of Education
3. TAXES The table shows the estimated sales tax revenues, in billions of dollars, for
Massachusetts each year since 2004.
Year
2004
2005
2006
2007
2008
Tax Revenue
3.75
3.89
4.00
4.17
4.47
Source: Beacon Hill Institute
4. PURCHASING The SureSave supermarket chain closely monitors how many diapers are
sold each year so that they can reasonably predict how many diapers will be sold in the
following year.
Year
2006
2007
2008
2009
2010
Diapers Sold
60,200
65,000
66,300
65,200
70,600
a. Find an equation for the median-fit line.
b. How many diapers should SureSave anticipate selling in 2011?
5. FARMING Some crops, such as barley, are very sensitive to how acidic the soil is. To
determine the ideal level of acidity, a farmer measured how many bushels of barley he
harvests in different fields with varying acidity levels.
Soil Acidity (pH)
5.7
6.2
6.6
6.8
7.1
Bushels Harvested
3
20
48
61
73
a. Find an equation for the regression line.
b. According to the equation, how many bushels would the farmer harvest if the soil had
a pH of 10?
c. Is this a reasonable prediction? Explain.
Skills Practice
Regression and Median-Fit Lines
4-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
40
Glencoe Algebra 1
Write an equation of the regression line for the data in each table below. Then
find the correlation coefficient.
1. TURTLES The table shows the number of turtles hatched at a zoo each year since 2006.
Year
2006
2007
2008
2009
2010
Turtles Hatched
21
17
16
16
14
2. SCHOOL LUNCHES The table shows the percentage of students receiving free or
reduced price school lunches at a certain school each year since 2006.
Year
2006
2007
2008
2009
2010
Percentage
14.4%
15.8%
18.3%
18.6%
20.9%
Source: KidsData
3. SPORTS Below is a table showing the number of students signed up to play lacrosse
after school in each age group.
Age
13
14
15
16
17
Lacrosse Players
17
14
6
9
12
4. LANGUAGE The State of California keeps track of how many millions of students are
learning English as a second language each year.
Year
2003
2004
2005
2006
2007
English Learners
1.600
1.599
1.592
1.570
1.569
Source: California Department of Education
a. Find an equation for the median-fit line.
b. Predict the number of students who were learning English in California in 2001.
c. Predict the number of students who were learning English in California in 2010.
5. POPULATION Detroit, Michigan, like a number of large cities, is losing population
every year. Below is a table showing the population of Detroit each decade.
Year
1960
1970
1980
1990
2000
Population (millions)
1.67
1.51
1.20
1.03
0.95
Source: U.S. Census Bureau
a. Find an equation for the regression line.
b. Find the correlation coefficient and explain the meaning of its sign.
c. Estimate the population of Detroit in 2008.
Practice
Regression and Median-Fit Lines
4-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-6
4-6
Chapter 4
41
Glencoe Algebra 1
Word Problem Practice
Regression and Median-Fit Lines
1. FOOTBALL Rutgers University running
back Ray Rice ran for 1732 total yards in
the 2007 regular season. The table below
shows his cumulative total number of
yards ran after select games.
Game
Number
1
3
6
9
12
Cumulative
Yards
184
431
818
1257 1732
Source: Rutgers University Athletics
Use a calculator to find an equation for
the regression line showing the total
yards y scored after x games. What is
the real-world meaning of the value
returned for a?
2. GOLD Ounces of gold are traded by
large investment banks in commodity
exchanges much the same way that
shares of stock are traded. The table
below shows the cost of a single ounce of
gold on the last day of trading in given
years.
Year
2002
2003
2004
2005
2006
Price
$346.70 $414.80
$438.10
$517.20
$636.30
Source: Global Financial Data
Use a calculator to find an equation for
the regression line. Then predict the
price of an ounce of gold on the last day
of trading in 2009. Is this a reasonable
prediction? Explain.
3. GOLF SCORES Emmanuel is practicing
golf as part of his school’s golf team.
Each week he plays a full round of golf
and records his total score. His scorecard
after five weeks is below.
Week
1
2
3
4
5
Golf Score
112
107
108
104
98
Use a calculator to find an equation for
the median-fit line. Then estimate how
many games Emmanuel will have to play
to get a score of 86.
4. STUDENT ELECTIONS The vote totals
for five of the candidates participating in
Montvale High School’s student council
elections and the number of hours each
candidate spent campaigning are shown
in the table below.
Hours
Campaigning
1
3
4
6
8
Votes Received
9
22
24
46
78
a. Use a calculator to find an equation
for the median-fit line.
b. Plot the data points and draw the
median-fit line on the graph below.
Votes Received
20
30
10
0
40
50
60
70
80
Campaign Time (h)
3
2
1
5
7
4
6
8 x
y
c. Suppose a sixth candidate spends
7 hours campaigning. Estimate how
many votes that candidate could
expect to receive.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
42
Glencoe Algebra 1
For some sets of data, a linear equation in the form y = ax + b does not adequately describe
the relationship between data points. The “QuadReg” function on a graphing calculator
will output an equation in the form y = ax2 + bx + c. The value of R2, the coefficient of
determination tells you how closely the parabola fits the data.
The table shows the population of Atlanta in various years.
Year
1970
1980
1990
2000
2005
2007
Population
497,000
425,000
394,017
416,474
470,688
498,109
Source: U.S. Census Bureau
a. Find the equation of a quadratic-regression parabola for the data.
Running a linear regression on the data provides an r value of 0.03, which indicates a
poor fit. The data appears to be a good candidate for a quadratic regression.
Step 1 Enter the data by pressing STAT and selecting the Edit
option. Enter the years since 1970 as your x-values (L1)
and enter the population figures as your y-values (L2).
Step 2 Perform the quadratic regression by pressing STAT and
selecting the CALC option. Scroll down to QuadReg and
press ENTER .
Step 3 Write the equation of the best-fit parabola by rounding the
a, b, and c values on the screen.
The equation for the best-fit parabola is
y = 302.8x2 – 11,480x + 501,227.
b. Find the coefficient of determination.
The coefficient of determination for the parabola is R2 = 0.969,
which indicates a good fit.
c. Use the quadratic-regression parabola to predict the population in 2010.
Graph the best-fit parabola. Then use the TRACE
feature and
the arrow keys until you find a point where x = 40.
When x ≈ 40, y ≈ 525,000. The estimated population will be 525,000.
Exercises
1. The table below shows the average high temperature in Crystal
River, Florida in various months.
Month
Jan (1)
Mar (3)
May (5)
Jul (7)
Sep (9)
Nov (11)
Avg. High (°F)
68°
76°
87°
91°
88°
76°
Source: Country Studies
a. Find the equation of the best-fit parabola.
b. Find the coefficient of determination.
c. Use the quadratic-regression parabola to predict the average high temperature in
April (4th month).
Enrichment
Quadratic Regression Parabolas
Example
4-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-7
Chapter 4
43
Glencoe Algebra 1
Inverse Relations An inverse relation is the set of ordered pairs obtained by
exchanging the x-coordinates with the y-coordinates of each ordered pair. The domain of a
relation becomes the range of its inverse, and the range of the relation becomes the domain
of its inverse.
Find and graph the inverse of the relation represented by line a.
The graph of the relation passes through (–2, –10), (–1, –7), (0, –4), (1, –1), (2, 2), (3, 5),
and (4, 8).
To find the inverse, exchange the coordinates
of the ordered pairs.
The graph of the inverse passes through the points
(–10, –2), (–7, –1), (–4, 0), (–1, 1), (2, 2), (5, 3), and (8, 4).
Graph these points and then draw the line that passes
through them.
Exercises
Find the inverse of each relation.
1. {(4, 7), (6, 2), (9, –1), (11, 3)}
2. {(–5, –9), (–4, –6), (–2, –4), (0, –3)}
3.
x
y
–8 –15
–2 –11
1
–8
5
1
11
8
4.
x
y
–8
3
–2
9
2
13
6
18
8
19
5.
x
y
–6
14
–5
11
–4
8
–3
5
–2
2
Graph the inverse of each relation.
6.
y
x
O
8
4
−4
−8
4
8
−4
−8
7.
y
x
O
8
4
−4
−8
4
8
−4
−8
8.
y
x
O
8
4
−4
−8
4
8
−4
−8
Study Guide
Inverse Linear Functions
Example
y
x
O
8
4
−4
−8
4
8
−4
−8
(−10, −2)
(−4, 0)
(2, 2)
(8, 4)
a
4-7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
44
Glencoe Algebra 1
Study Guide (continued)
Inverse Linear Functions
Inverse Functions A linear relation that is described by a function has an inverse
function that can generate ordered pairs of the inverse relation. The inverse of the linear
function f (x) can be written as f -1 (x) and is read f of x inverse or the inverse of f of x.
Find the inverse of f (x) = 3 −
4
x + 6.
Step 1 f (x) = 3 −
4
x + 6
Original equation
y = 3 −
4
x + 6
Replace f (x) with y.
Step 2 x = 3 −
4
y + 6
Interchange y and x.
Step 3 x - 6 = 3 −
4
y
Subtract 6 from each side.
4 −
3
(x - 6) = y
Multiply each side by 4 −
3
.
Step 4 4 −
3
(x - 6) = f -1 (x)
Replace y with f -1 (x).
The inverse of f (x) = 3 −
4
x + 6 is f -1 (x) = 4 −
3
(x - 6) or f -1 (x) = 4 −
3
x - 8.
Exercises
Find the inverse of each function.
1. f (x) = 4x - 3
2. f (x) = -3x + 7
3. f (x) = 3 −
2
x - 8
4. f (x) = 16 - 1 −
3
x
5. f (x) = 3(x - 5)
6. f (x) = -15 - 2 −
5
x
7. TOOLS Jimmy rents a chainsaw from the department store to work on his yard.
The total cost C(x) in dollars is given by C(x) = 9.99 + 3.00x, where x is the
number of days he rents the chainsaw.
a. Find the inverse function C -1 (x).
b. What do x and C -1 (x) represent in the context of the inverse function?
c. How many days did Jimmy rent the chainsaw if the total cost
was $27.99?
Example
4-7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-7
Chapter 4
45
Glencoe Algebra 1
Find the inverse of each relation.
1. x
y
–9
–1
–7
–4
–5
–7
–3 –10
–1 –13
2.
x
y
1
8
2
6
3
4
4
2
5
0
3.
x
y
–4
–2
–2
–1
0
1
2
0
4
2
4. {(-3, 2), (-1, 8), (1, 14), (3, 20)}
5. {(5, -3), (2, -9), (-1, -15), (-4, -21)}
6. {(4, 6), (3, 1), (2, -4), (1, -9)}
7. {(-1, 16), (-2, 12), (-3, 8), (-4, 4)}
Graph the inverse of each function.
8.
y
x
O
8
4
−4
−8
4
8
−4
−8
9.
y
x
O
8
4
−4
−8
4
8
−4
−8
10.
y
x
O
8
4
−4
−8
4
8
−4
−8
Find the inverse of each function.
11. f (x) = 8x - 5
12. f (x) = 6(x + 7)
13. f (x) = 3 −
4
x + 9
14. f (x) = -16 + 2 −
5
x
15. f (x) = 3x + 5 −
4
16. f (x) = -4x + 1 −
5
17. LEMONADE Chrissy spent $5.00 on supplies and lemonade powder for her lemonade
stand. She charges $0.50 per glass.
a. Write a function P(x) to represent her profit per glass sold.
b. Find the inverse function, P -1 (x).
c. What do x and P -1 (x) represent in the context of the inverse function?
d. How many glasses must Chrissy sell in order to make a $3 profit?
Skills Practice
Inverse Linear Functions
4-7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
46
Glencoe Algebra 1
Find the inverse of each relation.
1. {(-2, 1), (-5, 0), (-8, -1), (-11, 2)}
2. {(3, 5), (4, 8), (5, 11), (6, 14)}
3. {(5, 11), (1, 6), (-3, 1), (-7, -4)}
4. {(0, 3), (2, 3), (4, 3), (6, 3)}
Graph the inverse of each function.
5.
y
x
O
8
4
−4
−8
4
8
−4
−8
6.
y
x
O
8
4
−4
−8
4
8
−4
−8
7.
y
x
O
8
4
−4
−8
4
8
−4
−8
Find the inverse of each function.
8. f (x) = 6 −
5
x - 3
9. f (x) = 4x + 2 −
3
10. f (x) = 3x - 1 −
6
11. f (x) = 3(3x + 4)
12. f (x) = -5(-x - 6)
13. f (x) = 2x - 3 −
7
Write the inverse of each equation in f -1 (x) notation.
13. 4x + 6y = 24
14. -3y + 5x = 18
15. x + 5y = 12
16. 5x + 8y = 40
17. -4y - 3x = 15 + 2y
18. 2x - 3 = 4x + 5y
19. CHARITY Jenny is running in a charity event. One donor is paying an initial amount of
$20.00 plus an extra $5.00 for every mile that Jenny runs.
a. Write a function D(x) for the total donation for x miles run.
b. Find the inverse function, D -1 (x).
c. What do x and D -1 (x) represent in the context of the inverse function?
Practice
Inverse Linear Functions
4-7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-7
Chapter 4
47
Glencoe Algebra 1
Word Problem Practice
Inverse Linear Functions
1. BUSINESS Alisha started a baking
business. She spent $36 initially on
supplies and can make 5 dozen brownies
at a cost of $12. She charges her
customers $10 per dozen brownies.
a. Write a function C(x) to represent
Alisha’s total cost per dozen
brownies.
b. Write a function E(x) to represent
Alisha’s earnings per dozen brownies
sold.
c. Find P (x) = E(x) - C(x). What does
P (x) represent?
d. Find C -1 (x), E -1 (x), and P -1 (x).
e. How many dozen brownies does
Alisha need to sell in order to make
a profit?
2. GEOMETRY The area of the base of a
cylindrical water tank is 66 square feet.
The volume of water in the tank is
dependent on the height of the water h
and is represented by the function
V(h) = 66h. Find V -1 (h). What will the
height of the water be when the volume
reaches 2310 cubic feet?
3. SERVICE A technician is working on a
furnace. He is paid $150 per visit plus
$70 for every hour he works on the
furnace.
a. Write a function C(x) to represent the
total charge for every hour of
work.
b. Find the inverse function, C -1 (x).
c. How long did the technician work on
the furnace if the total charge was
$640?
4. FLOORING Kara is having baseboard
installed in her basement. The total
cost C(x) in dollars is given by
C(x) = 125 + 16x, where x is the
number of pieces of wood required
for the installation.
a. Find the inverse function C -1 (x).
b. If the total cost was $269 and each
piece of wood was 12 feet long, how
many total feet of wood were
used?
5. BOWLING Libby’s family went bowling
during a holiday special. The special cost
$40 for pizza, bowling shoes, and
unlimited drinks. Each game cost $2.
How many games did Libby bowl if the
total cost was $112 and the six family
members bowled an equal number of
games?
4-7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
48
Glencoe Algebra 1
In a function, there is exactly one output for every input. In other words, every element in
the domain pairs with exactly one element in the range. When a function is one-to-one,
each element of the domain pairs with exactly one unique element in the range. When a
function is onto, each element of the range corresponds to an element in the domain.
If a function is both one-to-one and onto, then the inverse is also a function.
Determine whether each relation is a function. If it is a function, determine
if it is one-to-one, onto, both, or neither.
1.
11
16
-3
4
3
6
9
12
2.
1
2
3
4
5
-3
-2
0
4
5
3.
3
6
9
12
15
10
5
0
-5
4.
4
2
7
11
6
1
-2
-4
7
5.
2
6
13
2
3
4
6
8
6.
3
1
-9
10
2
4
11
17
19
Determine whether the inverse of each function is also a function.
7.
y
x
O
8
4
−4
−8
4
8
−4
−8
8.
y
x
O
8
4
−4
−8
4
8
−4
−8
9.
y
x
O
8
4
−4
−8
4
8
−4
−8
Enrichment
One-to-One and Onto Functions
2
6
9
12
-1
3
5
8
9
one–to–one
-3
-2
-1
2
6
3
5
10
onto
5
7
9
10
-6
-11
-15
-19
both
4-7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Assessment
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
Chapter 4
49
Glencoe Algebra 1
Read each question. Then fill in the correct answer.
1. A
B
C
D
2. F
G
H
J
3. A
B
C
D
4. F
G
H
J
5. A
B
C
D
6. F
G
H
J
Multiple Choice
Student Recording Sheet
Use this recording sheet with pages 280–281 of the Student Edition.
Short Response/Gridded Response
Record your answer in the blank.
For gridded response questions, also enter your answer in the grid by writing
each number or symbol in a box. Then fill in the corresponding circle for that
number or symbol.
7.
8.
(grid in)
9.
10a.
10b.
11a.
11b.
11c.
8.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
.
.
.
.
.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
Extended Response
Record your answers for Question 12 on the back of this paper.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
50
Glencoe Algebra 1
4
Rubric for Scoring Extended Response Test
General Scoring Guidelines
•
If a student gives only a correct numerical answer to a problem but does not show how
he or she arrived at the answer, the student will be awarded only 1 credit. All extended
response questions require the student to show work.
•
A fully correct answer for a multiple-part question requires correct responses for all
parts of the question. For example, if a question has three parts, the correct response to
one or two parts of the question that required work to be shown is not considered a fully
correct response.
•
Students who use trial and error to solve a problem must show their method. Merely
showing that the answer checks or is correct is not considered a complete response for
full credit.
Exercise 12 Rubric
Score
Specifi c Criteria
4
Student explain that the slopes of the lines must be compared. If two lines have
the same slope, they are parallel. If their slopes are opposite reciprocals, they are
perpendicular.
3
A generally correct solution, but may contain minor flaws in reasoning
or computation.
2
A partially correct interpretation and/or solution to the problem.
1
A correct solution with no evidence or explanation.
0
An incorrect solution indicating no mathematical understanding of the concept or
task, or no solution is given.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Assessment
NAME
DATE
PERIOD
Chapter 4
51
Glencoe Algebra 1
4
4
SCORE
For Questions 1 and 2, write an equation in slope-intercept
form for each situation.
1. slope: 1 −
4
, y-intercept: -5
2. line passing through (9, 2) and (-2, 6)
3. Graph 4x + 3y = 12.
4. Write a linear equation in slope-intercept form to model a
tree 4 feet tall that grows 3 inches per year.
5. MULTIPLE CHOICE The table of ordered pairs
shows the coordinates of the two points on
the graph of a function. Which equation
describes the function?
A y = -2x + 1
C y = - 1 −
2
x + 1
B y = 1 −
2
x - 1
D y = - 1 −
2
x - 1
Chapter 4 Quiz 2
(Lessons 4-3 and 4-4)
Chapter 4 Quiz 1
(Lessons 4-1 and 4-2)
x
y
-2
2
4 -1
1. Write an equation in point-slope form for a line that
passes through (3, 6) with a slope of - 1 −
3
.
2. Write y - 9 = -(x + 2) in slope-intercept form.
3. Write an equation in point-slope form for a horizontal
line that passes through (-4, -1).
4. Write an equation in slope-intercept form for the line
that passes through (5, 3) and is parallel to x + 3y = 6.
5. MULTIPLE CHOICE Line DE contains the points D (-1, -4) and
E (3, 3). Line FG contains the point F (-3, 3). Which set of
coordinates for point G makes the two lines perpendicular?
A (1, 7)
C (1, 4)
B (1, 10)
D (4, -1)
SCORE
1.
2.
3.
y
x
O
4.
5.
1.
2.
3.
4.
5.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NAME
DATE
PERIOD
Chapter 4
52
Glencoe Algebra 1
4
4
SCORE
Chapter 4 Quiz 3
(Lessons 4-5 and 4-6)
1. Find the inverse of {(1, 3), (4, -1), (7, -5), (10, -9)}.
2. Graph the inverse of the function
graphed at the right.
Find the inverse of each function.
3. f (x) = 4x + 6
4. f (x) = 3 −
4
x - 8
5. MULTIPLE CHOICE Write the inverse of
3x + 4y = 12 in f -1 (x) notation.
A f -1 (x) = 12 - 4x −
3
B f -1 (x) = 12 - 3x −
4
C f -1 (x) = 12 - 3x
D f -1 (x) =
12 - 4y
−
3
Chapter 4 Quiz 4
(Lesson 4-7)
For Questions 1–5, use the table.
Age (years)
26
27
28
29
30
Median Income
($1000)
16.8 19.1 23.3 25.8 33.9
1. Make a scatter plot relating age to median income.
Then draw a fit line for the scatter plot.
2. Determine whether the graph shows a positive correlation,
a negative correlation, or no correlation. If there is a positive
or negative correlation, describe its meaning.
3. Write an equation of the best-fit line for the data in the table.
4. Use the line of fit to predict the median income for
32-year olds.
5. MULTIPLE CHOICE What is the correlation coefficient for
the best-fit line?
A 4.09
B –90.74
C 0.943
D 0.971
SCORE
1.
2.
3.
4.
5.
Age (years)
Median Income ($1000)
16
19
22
25
28
31
26
0
27 28 29 30
y
x
O
8
4
−4
−8
4
8
−4
−8
1.
2.
3.
4.
5.
y
x
O
8
4
−4
−8
4
8
−4
−8
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Assessment
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
53
Glencoe Algebra 1
4
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. Which is the slope-intercept form of an equation for the line containing
(0, -3) with slope -1?
A y = -x - 3
B y = -3x - 1
C y = x + 3
D x = -3y - 1
2. Write an equation in slope-intercept form of the line with a slope of - 3 −
4
and y-intercept of – 5.
F y = 5x - 3 −
4
G 3x + 4y = 20
H y = - 3 −
4
x - 5
J y = - 3 −
4
x + 5
3. Write an equation of the line that passes through (-2, 8) and (-4, -4).
A y = 2x + 12
B y = 6x + 20
C y = -6x - 4
D y = 1 −
6
x + 25 −
3
4. Write y - 3 = 2 −
3
(x - 2) in standard form.
F 2x - 3y = 5
G y = 2 −
3
x + 5 −
3
H -2x + 3y = -5 J 2x - 3y = -5
5. Write y - 1 = 2 (x -
3 −
2
) in slope-intercept form.
A 2x - y = 2
B 1 −
2
y + 1 −
2
= x
C y = 2x - 1 −
2
D y = 2x - 2
6. A cell phone company charges $42 per month of service. The cost of a new
cell phone, plus 8 months of service, is $415.99. How much does it cost to
buy a new cell phone and 3 months of service?
F $79.99
G $126.00
H $205.99
J $289.99
Part II
For Questions 7–10, use the following information.
Nikko needs to get his air-conditioner fixed. The technician will charge Nikko a flat fee of
$50 plus an additional $20 for each hour of work.
7. Write an equation to represent Nikko’s total cost to repair
his air-conditioner. Use t for total cost and h for hours.
8. Graph this equation.
9. How much will it cost Nikko if the technician
has to spend 4 hours working on the air-conditioner?
10. How many hours must the technician work for it to
cost Nikko $180?
1.
2.
3.
4.
5.
6.
Assessment
Chapter 4 Mid-Chapter Test
(Lessons 4-1 through 4-3)
Part I
7.
8.
9.
10.
1
Hours
2 3 4 5 6 7 8 9 10
20
40
60
80
100
Total Cost ($)
120
140
160
180
200
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
54
Glencoe Algebra 1
4
SCORE
Choose from the terms above to complete each sentence.
1. If two lines have slopes that are negative reciprocals of each
other, then they are
.
2. A(n)
is the set of ordered pairs obtained
by exchanging the x-coordinates with the y-coordinates of
each ordered pair of a relation or function.
3. A graph of data points is sometimes called a
.
4. If two lines have slopes that are the same, then they are
.
5. The number that describes how closely a best-fit line models a
set of data is called the
.
6. The leftmost data point in a set is (3, 27) and the rightmost
point is (12, 13). If you use a linear prediction equation to find
the corresponding y-value for x = 10, you are using a method
called
.
7. The leftmost data point in a set is (1997, 24) and the rightmost
point is (2011, 38). If you use a linear prediction equation to
find the corresponding y-value for x = 2012, you are using a
method called
.
8. The equation y = -3x + 12 is written in
form.
9. The equation y + 6 = 2(x - 4) is written in
form.
Define each term in your own words.
10. line of fit
11. linear extrapolation
Chapter 4 Vocabulary Test
best-fi t line
bivariate data
correlation coefficient
inverse function
inverse relation
linear extrapolation
linear interpolation
linear regression
median-fi t line
line of fi t
parallel lines
perpendicular lines
point-slope form
rate of change
scatter plot
slope-intercept form
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Assessment
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
55
Glencoe Algebra 1
4
SCORE
Chapter 4 Test, Form 1
Write the letter for the correct answer in the blank at the right of each question.
For Questions 1–5, find the equation in slope-intercept form that
describes each line.
1. a line with slope -2 and y-intercept 4
A y = -2x
B y = 4x - 2
C y = -2x + 4
D y = 2x - 4
2. a line through (2, 4) with slope 0
F y = 2
G x = 2
H y = 4
J x = 4
3. a line through (4, 2) with slope 1 −
2
A y = - 1 −
2
x
B y = 1 −
2
x - 4
C y = 2x - 10
D y = 1 −
2
x
4. a line through (-1, 1) and (2, 3)
F y = 2 −
3
x + 5 −
3
G y = - 2 −
3
x + 5 −
3
H y = 2 −
3
x - 5 −
3
J y = - 2 −
3
x - 5 −
3
5. the line graphed at the right
A y = 2 −
3
x - 1
C y = 2 −
3
x + 3 −
2
B y = 3 −
2
x - 1
D y = 3 −
2
x + 3 −
2
6. If 5 deli sandwiches cost $29.75, how much will 8 sandwiches cost?
F $37.75
G $29.75
H $47.60
J $0.16
7. What is the standard form of y - 8 = 2(x + 3)?
A 2x + y = 14
B y = 2x + 14
C 2x - y = -14
D y - 2x = 11
8. Which is the graph of 3x - 4y = 6 ?
F
y
x
O
G
y
x
O
H
y
x
O
J
y
x
O
9. Which is the point-slope form of an equation for the line that passes through
(0, -5) with slope 2?
A y = 2x - 5
B y + 5 = 2x
C y - 5 = x - 2 D y = 2(x + 5)
10. What is the slope-intercept form of y + 6 = 2(x + 2)?
F y = 2x - 6
G y = 2x - 2
H y = 2x + 6
J 2x - y = 6
11. When are two lines parallel?
A when the slopes are opposite
B when the slopes are equal
C when the slopes are positive
D when the product of the slopes is -1
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
y
x
O
(3, 1)
(0, -1)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
56
Glencoe Algebra 1
4
Chapter 4 Test, Form 1 (continued)
12. Find the slope-intercept form of an equation for the line that passes
through (-1, 2) and is parallel to y = 2x - 3.
F y = 2x + 4
G y = 0.5x + 4
H y = 2x + 3
J y = -0.5x - 4
13. Find the slope-intercept form of an equation of the line perpendicular to
the graph of x - 3y = 5 and passing through (0, 6).
A y = 1 −
3
x - 2
B y = -3x + 6
C y = 1 −
3
x + 2
D y = 3x - 6
For Questions 14 and 15, use the scatter plot shown.
14. How would you describe the relationship between
the x- and y-values in the scatter plot?
F strong negative correlation
G weak negative correlation
H weak positive correlation
J strong positive correlation
15. Based on the data in the scatter plot, what
would you expect the y-value to be for x = 2020?
A greater than 80
C between 65 and 50
B between 80 and 65
D less than 50
16. Which equation has a slope of 2 and a y-intercept of -5?
F y = -5x + 2
G y = 5x + 2
H y = 2x + 5
J y = 2x - 5
17. Which correlation coefficient corresponds to the best-fit line that most closely
models its set of data?
A 0.84
B 0.13
C -0.87
D -0.15
18. The table below shows Mia’s bowling score each week she participated in a
bowling league.
Week
1
2
3
4
5
6
Score
122 131 130 133 145 139
Use the median-fit line to estimate Mia’s score for week 16.
F 173
G 180
H 182
J 257
19. If f(x) = 6x + 3, find f -1 (x).
A f -1 (x) = 6x - 3 B f -1 (x) = x - 6 −
3
C f -1 (x) = x - 3 −
6
D f -1 (x) = -3 - 6x
20. If f(x) = 4(3x - 5), find f -1 (x).
F f -1 (x) = x + 5 −
12
G f -1 (x) = x + 20 −
12
H f -1 (x) = x - 20 −
12
J f -1 (x) = x + 5 −
4
Bonus Find the value of r in (4, r), (r, 2) so that the slope of the
line containing them is - 5 −
3
.
0
50
60
70
80
90
'90 '95 '00 '05 '10
B:
12.
13.
14.
15.
16.
17.
18.
19.
20.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Assessment
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
57
Glencoe Algebra 1
4
SCORE
Assessment
Write the letter for the correct answer in the blank at the right of each question.
1. What is the slope-intercept form of the equation of a line with a slope of 5
and a y-intercept of -8?
A y = -8x + 5
B y = 8x - 5
C 5x - y = - 8
D y = 5x - 8
2. Which equation is graphed at the right?
F 2y - x =10
H 2x - y = 5
G 2x + y = -5
J 2y + x = -5
3. Which is an equation of the line that passes
through (2, -5) and (6, 3)?
A y = 1 −
2
x - 6
C y = 2x + 12
B y = 1 −
2
x
D y = 2x - 9
4. What is an equation of the line through (0, -3) with slope 2 −
5
?
F -5x + 2y = 15
H 2x - 5y = 15
G -5x - 2y = -15
J -2x + 5y = 15
5. Which is an equation of the line with slope -3 and a y-intercept of 5?
A y = -3(x + 5)
B y - 5 = -3x
C -3x + y = 5
D y = 5x - 3
6. What is the equation of the line through (-2, -3) with a slope of 0?
F x = -2
G y = -3
H -2x - 3y = 0
J -3x + 2y = 0
7. Find the slope-intercept form of the equation of the line that passes through
(-5, 3) and is parallel to 12x - 3y = 10.
A y = -4x - 17
B y = 4x - 13
C y = - 4x + 13 D y = 4x + 23
8. If line q has a slope of - 3 −
8
, what is the slope of any line perpendicular to q?
F - 3 −
8
G 3 −
8
H 8 −
3
J - 8 −
3
9. A line of fit might be defined as
A a line that connects all the data points.
B a line that might best estimate the data and be used for predicting values.
C a vertical line halfway through the data.
D a line that has a slope greater than 1.
10. A scatter plot of data comparing the number of years since Holbrook High
School introduced a math club and the number of students participating
contains the ordered pairs (3, 19) and (8, 42). Which is the slope-intercept
form of an equation for the line of fit?
F y = 4.6x + 5.2
G y = 3x + 1
H y = 5.2x + 4.6
J y = 0.22x - 1.13
11. Use the equation from Question 10 to estimate the number of students who
will be in the math club during the 15th year.
A 53
B 61
C 65
D 74
Chapter 4 Test, Form 2A
y
x
O
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
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CONSUMABLE WORKBOOKS Many of the worksheets contained in the Chapter Resource Masters
booklets are available as consumable workbooks in both English and Spanish.
MHID
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Study Guide and Intervention Workbook
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978-0-07-660292-6
Homework Practice Workbook
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Spanish Version
Homework Practice Workbook
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Answers For Workbooks The answers for Chapter 4 of these workbooks can be found in the
back of this Chapter Resource Masters booklet.
ConnectED All of the materials found in this booklet are included for viewing, printing, and editing at
connected.mcgraw-hill.com.
Spanish Assessment Masters (MHID: 0-07-660289-3, ISBN: 978-0-07-660289-6) These masters
contain a Spanish version of Chapter 4 Test Form 2A and Form 2C.
iii
Teacher’s Guide to Using the Chapter 4
Resource Masters ..............................................iv
Chapter Resources
Chapter 4 Student-Built Glossary ...................... 1
Chapter 4 Anticipation Guide (English) ............. 3
Chapter 4 Anticipation Guide (Spanish) ............ 4
Lesson 4-1
Graphing Equations in Slope-Intercept Form
Study Guide and Intervention ............................ 5
Skills Practice .................................................... 7
Practice ............................................................. 8
Word Problem Practice ...................................... 9
Enrichment ...................................................... 10
Lesson 4-2
Writing Equations in Slope-Intercept Form
Study Guide and Intervention ...........................11
Skills Practice .................................................. 13
Practice ........................................................... 14
Word Problem Practice .................................... 15
Enrichment ...................................................... 16
Lesson 4-3
Writing Equations in Point-Slope Form
Study Guide and Intervention .......................... 17
Skills Practice .................................................. 19
Practice ........................................................... 20
Word Problem Practice .................................... 21
Enrichment ...................................................... 22
Graphing Calculator Activity ............................ 23
Lesson 4-4
Parallel and Perpendicular Lines
Study Guide and Intervention .......................... 24
Skills Practice .................................................. 26
Practice ........................................................... 27
Word Problem Practice .................................... 28
Enrichment ...................................................... 29
Lesson 4-5
Scatter Plots and Lines of Fit
Study Guide and Intervention .......................... 30
Skills Practice .................................................. 32
Practice ........................................................... 33
Word Problem Practice .................................... 34
Enrichment ...................................................... 35
Spreadsheet Activity ........................................ 36
Lesson 4-6
Regression and Median-Fit Lines
Study Guide and Intervention .......................... 37
Skills Practice .................................................. 39
Practice ........................................................... 40
Word Problem Practice .................................... 41
Enrichment ...................................................... 42
Lesson 4-7
Inverse Linear Functions
Study Guide and Intervention .......................... 43
Skills Practice .................................................. 45
Practice ........................................................... 46
Word Problem Practice .................................... 47
Enrichment ...................................................... 48
Assessment
Student Recording Sheet ................................ 49
Rubric for Scoring Extended Response .......... 50
Chapter 4 Quizzes 1 and 2 ............................. 51
Chapter 4 Quizzes 3 and 4 ............................. 52
Chapter 4 Mid-Chapter Test ............................ 53
Chapter 4 Vocabulary Test ............................... 54
Chapter 4 Test, Form 1 .................................... 55
Chapter 4 Test, Form 2A ................................. 57
Chapter 4 Test, Form 2B ................................. 59
Chapter 4 Test Form 2C .................................. 61
Chapter 4 Test Form 2D .................................. 63
Chapter 4 Test Form 3 ..................................... 65
Chapter 4 Extended-Response Test ................ 67
Standardized Test Practice .............................. 68
Answers ........................................... A1–A34
Contents
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
iv
Teacher’s Guide to Using the
Chapter 4 Resource Masters
The Chapter 4 Resource Masters includes the core materials needed for Chapter 4. These
materials include worksheets, extensions, and assessment options. The answers for these
pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing, printing, and
editing at connectED.mcgraw-hill.com.
Chapter Resources
Student-Built Glossary (pages 1–2) These
masters are a student study tool that
presents up to twenty of the key vocabulary
terms from the chapter. Students are to
record definitions and/or examples for each
term. You may suggest that students
highlight or star the terms with which they
are not familiar. Give this to students before
beginning Lesson 4-1. Encourage them to
add these pages to their mathematics study
notebooks. Remind them to complete the
appropriate words as they study each lesson.
Anticipation Guide (pages 3–4) This
master, presented in both English and
Spanish, is a survey used before beginning
the chapter to pinpoint what students may
or may not know about the concepts in the
chapter. Students will revisit this survey
after they complete the chapter to see if
their perceptions have changed.
Lesson Resources
Study Guide and Intervention These
masters provide vocabulary, key concepts,
additional worked-out examples and Check
Your Progress exercises to use as a
reteaching activity. It can also be used in
conjunction with the Student Edition as an
instructional tool for students who have
been absent.
Skills Practice This master focuses more
on the computational nature of the lesson.
Use as an additional practice option or as
homework for second-day teaching of the
lesson.
Practice This master closely follows the
types of problems found in the Exercises
section of the Student Edition and includes
word problems. Use as an additional
practice option or as homework for second-
day teaching of the lesson.
Word Problem Practice This master
includes additional practice in solving word
problems that apply the concepts of the
lesson. Use as an additional practice or as
homework for second-day teaching of the
lesson.
Enrichment These activities may extend
the concepts of the lesson, offer an historical
or multicultural look at the concepts, or
widen students’ perspectives on the
mathematics they are learning. They are
written for use with all levels of students.
Graphing Calculator, TI-Nspire, or
Spreadsheet Activities
These activities present ways in which
technology can be used with the concepts in
some lessons of this chapter. Use as an
alternative approach to some concepts or as
an integral part of your lesson presentation.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
v
Assessment Options
The assessment masters in the Chapter 4
Resource Masters offer a wide range
of assessment tools for formative
(monitoring) assessment and summative
(final) assessment.
Student Recording Sheet This master
corresponds with the standardized test
practice at the end of the chapter.
Extended Response Rubric This master
provides information for teachers and stu-
dents on how to assess performance on open-
ended questions.
Quizzes Four free-response quizzes offer
assessment at appropriate intervals in
the chapter.
Mid-Chapter Test This 1-page test
provides an option to assess the first half of
the chapter. It parallels the timing of the
Mid-Chapter Quiz in the Student Edition
and includes both multiple-choice and free-
response questions.
Vocabulary Test This test is suitable for
all students. It includes a list of vocabulary
words and 11 questions to assess students’
knowledge of those words. This can also be
used in conjunction with one of the leveled
chapter tests.
Leveled Chapter Tests
• Form 1 contains multiple-choice ques-
tions and is intended for use with below
grade level students.
• Forms 2A and 2B contain multiple-
choice questions aimed at on grade level
students. These tests are similar in
format to offer comparable testing
situations.
• Forms 2C and 2D contain free-
response questions aimed at on grade
level students. These tests are similar in
format to offer comparable testing
situations.
• Form 3 is a free-response test for use
with above grade level students.
All of the above mentioned tests include a
free-response Bonus question.
Extended-Response Test Performance
assessment tasks are suitable for all
students. Sample answers and a scoring
rubric are included for evaluation.
Standardized Test Practice These three
pages are cumulative in nature. It includes
three parts: multiple-choice questions with
bubble-in answer format, griddable
questions with answer grids, and short-
answer free-response questions.
Answers
• The answers for the Anticipation Guide
and Lesson Resources are provided as
reduced pages.
• Full-size answer keys are provided for
the assessment masters.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter Resources
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
Chapter 4
1
Glencoe Algebra 1
Student-Built Glossary
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 4.
As you study the chapter, complete each term’s definition or description.
Remember to add the page number where you found the term. Add these pages to
your Algebra Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term
Found
on Page
Defi nition/Description/Example
best-fit line
bivariate data
correlation coefficient
kawr·uh·LAY·shun
inverse function
inverse relation
line of fit
linear extrapolation
ihk·STRA·puh·LAY·shun
linear interpolation
ihn·TUHR·puh·LAY·shun
(continued on the next page)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
Chapter 4
2
Glencoe Algebra 1
Student-Built Glossary (continued)
Vocabulary Term
Found
on Page
Defi nition/Description/Example
linear regression
median-fit line
parallel lines
perpendicular lines
PUHR·puhn·DIH·kyuh·luhr
residual
scatter plot
slope-intercept form
IHN·tuhr·SEHPT
Chapter Resources
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
Chapter 4
3
Glencoe Algebra 1
Anticipation Guide
Equations of Linear Functions
Before you begin Chapter 4
• Read each statement.
• Decide whether you Agree (A) or Disagree (D) with the statement.
• Write A or D in the first column OR if you are not sure whether you agree or
disagree, write NS (Not Sure).
After you complete Chapter 4
• Reread each statement and complete the last column by entering an A or a D.
• Did any of your opinions about the statements change from the first column?
• For those statements that you mark with a D, use a piece of paper to write an
example of why you disagree.
STEP 1
A, D, or NS
Statement
STEP 2
A or D
1. The slope of a line given by an equation in the form y = mx + b
can be determined by looking at the equation.
2. The y-intercept of y = 12x - 8 is 8.
3. If two points on a line are known, then an equation can be
written for that line.
4. An equation in the form y = mx + b is in point-slope form.
5. If a pair of lines are parallel, then they have the same slope.
6. Lines that intersect at right angles are called perpendicular
lines.
7. A scatter plot is said to have a negative correlation when the
points are random and show no relationship between x and y.
8. The closer the correlation coefficient is to zero, the more closely
a best-fit line models a set of data.
9. The equations of a regression line and a median-fit line are
very similar.
10. An inverse relation is obtained by exchanging the x-coordinates
with the y-coordinates of each ordered pair of the original
relation.
Step 1
Step 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Capítulo 4
4
Álgebra 1 de Glencoe
4
Ejercicios preparatorios
Ecuaciones de Funciones Lineales
Antes de comenzar el Capítulo 4
• Lee cada enunciado.
• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.
• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta,
escribe NS (No estoy seguro(a)).
Después de completar el Capítulo 4
• Vuelve a leer cada enunciado y completa la última columna con una A o una D.
• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?
• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los
enunciados que marcaste con una D.
PASO 1
A, D, o NS
Enunciado
PASO 2
A o D
1. La pendiente de una recta dada por una ecuación de la forma
y = mx + b se puede determinar mediante la observación de
la ecuación.
2. La intersección y de y = 12x - 8 es 8.
3. Si se conocen dos puntos sobre una recta, entonces se puede
escribir una ecuación para esa recta.
4. Una ecuación de la forma y = mx + b está en forma
punto-pendiente.
6. A las rectas que se intersecan en ángulos rectos se les
llama rectas perpendiculares.
7. Se dice que un diagrama de dispersión tiene correlación negativa
cuando los puntos son aleatorios y no muestran relación entre x y y.
8. Entre más cercano se encuentre de cero el coeficiente de
correlación, mejor modela un conjunto de datos la recta de
mejor ajuste.
9. La ecuación de una línea de regresión y una recta de mediano
ajuste son muy parecidas.
10. Una relación inversa es obtenida cambiando las x-coordenadas
con las y-coordenadas de cada par pedido de la relación original.
Paso 1
Paso 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-1
4-1
Chapter 4
5
Glencoe Algebra 1
Study Guide and Intervention
Graphing Equations in Slope-Intercept Form
Slope-Intercept Form
Slope-Intercept Form
y = mx + b, where m is the slope and b is the y-intercept
Write an equation in slope-intercept form for the line with a slope
of -4 and a y-intercept of 3.
y = mx + b
Slope-intercept form
y = -4x + 3
Replace m with -4 and b with 3.
Graph 3x - 4y = 8.
3x - 4y = 8
Original equation
-4y = -3x + 8
Subtract 3x from each side.
-4y
−
-4
= -3x + 8 −
-4
Divide each side by -4.
y = 3 −
4
x - 2
Simplify.
The y-intercept of y = 3 −
4
x - 2 is -2 and the slope is 3 −
4
. So graph the point (0, -2). From
this point, move up 3 units and right 4 units. Draw a line passing through both points.
Exercises
Write an equation of a line in slope-intercept form with the given slope and
y-intercept.
1. slope: 8, y-intercept -3
2. slope: -2, y-intercept -1
3. slope: -1, y-intercept -7
Write an equation in slope-intercept form for each graph shown.
4.
(0, –2)
(1, 0)
x
y
O
5.
(3, 0)
(0, 3)
x
y
O
6.
(4, –2)
(0, –5)
x
y
O
Graph each equation.
7. y = 2x + 1
8. y = -3x + 2
9. y = -x - 1
x
y
O
x
y
O
x
y
O
(0, –2)
(4, 1)
x
y
O
3x - 4y = 8
Example 1
Example 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4-1
Chapter 4
6
Glencoe Algebra 1
Study Guide and Intervention (continued)
Graphing Equations in Slope-Intercept Form
Modeling Real-World Data
MEDIA Since 1999, the number of music cassettes sold has
decreased by an average rate of 27 million per year. There were 124 million music
cassettes sold in 1999.
a. Write a linear equation to find the average number of music cassettes sold in
any year after 1999.
The rate of change is -27 million per year. In the first year, the number of music
cassettes sold was 124 million. Let N = the number of millions of music cassettes sold.
Let x = the number of years since 1999. An equation is N = -27x + 124.
b. Graph the equation.
The graph of N = -27x + 124 is a line that passes
through the point at (0, 124) and has a slope of -27.
c. Find the approximate number of music cassettes
sold in 2003.
N = -27x + 124
Original equation
N = -27(4) + 124
Replace x with 4.
N = 16
Simplify.
There were about 16 million music cassettes sold in 2003.
Exercises
1. MUSIC In 2001, full-length cassettes represented 3.4% of
total music sales. Between 2001 and 2006, the percent
decreased by about 0.5% per year.
a. Write an equation to find the percent P of recorded music
sold as full-length cassettes for any year x between
2001 and 2006.
b. Graph the equation on the grid at the right.
c. Find the percent of recorded music sold
as full-length cassettes in 2004.
2. POPULATION The population of the United States is
projected to be 300 million by the year 2010. Between
2010 and 2050, the population is expected to increase
by about 2.5 million per year.
a. Write an equation to find the population P in any year x
between 2010 and 2050.
b. Graph the equation on the grid at the right.
c. Find the population in 2050.
Full-length Cassette Sales
Percent of Total Music Sales
1.5%
2.0%
1.0%
2.5%
3.0%
3.5%
Years Since 2001
3
2
1
0
5
4
Projected United
States Population
Years Since 2010
Population (millions)
0
20
40
x
P
400
380
360
340
320
300
Music Cassettes Sold
Cassettes (millions)
50
75
25
0
100
125
Years Since 1999
3
2
1
5
7
4
6
Example
Lesson X-1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-1
4-1
Chapter 4
7
Glencoe Algebra 1
Skills Practice
Graphing Equations in Slope-Intercept Form
Write an equation of a line in slope-intercept form with the given slope
and y-intercept.
1. slope: 5, y-intercept: -3
2. slope: -2, y-intercept: 7
3. slope: -6, y-intercept: -2
4. slope: 7, y-intercept: 1
5. slope: 3, y-intercept: 2
6. slope: -4, y-intercept: -9
7. slope: 1, y-intercept: -12
8. slope: 0, y-intercept: 8
Write an equation in slope-intercept form for each graph shown.
9.
(2, 1)
(0, –3)
x
y
O
10.
(0, 2)
(2, –4)
x
y
O
11.
(0, –1)
(2, –3)
x
y
O
Graph each equation.
12. y = x + 4
13. y = -2x - 1
14. x + y = -3
x
y
O
x
y
O
x
y
O
15. VIDEO RENTALS A video store charges $10 for a rental card
plus $2 per rental.
a. Write an equation in slope-intercept form for the total
cost c of buying a rental card and renting m
movies.
b. Graph the equation.
c. Find the cost of buying a rental card and renting 6 movies.
Video Store
Rental Costs
Total Cost ($)
10
0
12
14
16
18
20
c
Movies Rented
1
2
3
4
5 m
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4-1
Chapter 4
8
Glencoe Algebra 1
Practice
Graphing Equations in Slope-Intercept Form
Write an equation of a line in slope-intercept form with the given slope and
y-intercept.
1. slope: 1 −
4
, y-intercept: 3
2. slope: 3 −
2
, y-intercept: -4
3. slope: 1.5, y-intercept: -1
4. slope: -2.5, y-intercept: 3.5
Write an equation in slope-intercept form for each graph shown.
5.
(–5, 0)
(0, 2)
x
y
O
6.
(–2, 0)
(0, 3)
x
y
O
7.
(–3, 0)
(0, –2)
x
y
O
Graph each equation.
8. y = - 1 −
2
x + 2
9. 3y = 2x - 6
10. 6x + 3y = 6
x
y
O
x
y
O
x
y
O
11. WRITING Carla has already written 10 pages of a novel.
She plans to write 15 additional pages per month until she
is finished.
a. Write an equation to find the total number of pages P
written after any number of months m.
b. Graph the equation on the grid at the right.
c. Find the total number of pages written after 5 months.
Carla’s Novel
Months
Pages Written
2
0
4
6
1
3
5
m
P
100
80
60
40
20
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-1
4-1
Chapter 4
9
Glencoe Algebra 1
Word Problem Practice
Graphing Equations in Slope-Intercept Form
1. SAVINGS Wade’s grandmother gave him
$100 for his birthday. Wade wants to
save his money to buy a new MP3 player
that costs $275. Each month, he adds
$25 to his MP3 savings. Write an
equation in slope-intercept form for x, the
number of months that it will take Wade
to save $275.
2. CAR CARE Suppose regular gasoline
costs $2.76 per gallon. You can purchase
a car wash at the gas station for $3. The
graph of the equation for the cost of x
gallons of gasoline and a car wash is
shown below. Write the equation in slope-
intercept form for the line.
Gasoline (gal)
3
2
1
0
5
4
9
8
7
10
y
x
6
Cost of gas and car wash ($)
6
8
4
2
10
16
14
12
18
24
22
20
(4, 14.04)
(2, 8.52)
(0, 3)
3. ADULT EDUCATION Angie’s mother
wants to take some adult education
classes at the local high school. She has
to pay a one-time enrollment fee of $25
to join the adult education community,
and then $45 for each class she wants to
take. The equation y = 45x + 25
expresses the cost of taking x classes.
What are the slope and y-intercept of
the equation?
4. BUSINESS A construction crew needs to
rent a trench digger for up to a week. An
equipment rental company charges $40
per day plus a $20 non-refundable
insurance cost to rent a trench digger.
Write and graph an equation to find the
total cost to rent the trench digger for
d days.
Days
3
2
1
0
5
4
9
8
7
6
Price ($)
100
140
60
20
180
300
340
260
220
5. ENERGY From 2002 to 2005, U.S.
consumption of renewable energy
increased an average of 0.17 quadrillion
BTUs per year. About 6.07 quadrillion
BTUs of renewable power were produced
in the year 2002.
a. Write an equation in slope-intercept
form to find the amount of renewable
power P (quadrillion BTUs) produced
in year y between 2002 and 2005.
b. Approximately how much renewable
power was produced in 2005?
c. If the same trend continues from 2006
to 2010, how much renewable power
will be produced in the year 2010?
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4-1
Chapter 4
10
Glencoe Algebra 1
Enrichment
Using Equations: Ideal Weight
You can find your ideal weight as follows.
A woman should weigh 100 pounds for the first 5 feet of height and
5 additional pounds for each inch over 5 feet (5 feet = 60 inches).
A man should weigh 106 pounds for the first 5 feet of height and
6 additional pounds for each inch over 5 feet. These formulas apply to
people with normal bone structures.
To determine your bone structure, wrap your thumb and index finger
around the wrist of your other hand. If the thumb and finger just touch,
you have normal bone structure. If they overlap, you are small-boned.
If they don’t overlap, you are large-boned. Small-boned people should decrease their
calculated ideal weight by 10%. Large-boned people should increase the value by 10%.
Calculate the ideal weights of these people.
1. woman, 5 ft 4 in., normal-boned
2. man, 5 ft 11 in., large-boned
3. man, 6 ft 5 in., small-boned
4. you, if you are at least 5 ft tall
For Exercises 5–9, use the following information.
Suppose a normal-boned man is x inches tall. If he is at least 5 feet
tall, then x - 60 represents the number of inches this man is over
5 feet tall. For each of these inches, his ideal weight is increased by
6 pounds. Thus, his proper weight y is given by the formula
y = 6(x - 60) + 106 or y = 6x - 254. If the man is large-boned, the
formula becomes y = 6x - 254 + 0.10(6x - 254).
5. Write the formula for the weight of a large-boned man
in slope-intercept form.
6. Derive the formula for the ideal weight y of a normal-boned
female with height x inches. Write the formula in
slope-intercept form.
7. Derive the formula in slope-intercept form for the ideal weight y
of a large-boned female with height x inches.
8. Derive the formula in slope-intercept form for the ideal weight y
of a small-boned male with height x inches.
9. Find the heights at which the ideal weights of normal-boned males
and large-boned females would be the same.
Lesson X-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-2
4-2
Chapter 4
11
Glencoe Algebra 1
Exercises
Write an equation of the line that passes through the given point and has the
given slope.
1.
(3, 5)
x
y
O
m = 2
2.
(0, 0)
x
y
O
m = –2
3.
(2, 4)
x
y
O
m = 12
4. (8, 2); slope - 3 −
4
5. (-1, -3); slope 5
6. (4, -5); slope - 1 −
2
7. (-5, 4); slope 0
8. (2, 2); slope 1 −
2
9. (1, -4); slope -6
10. (-3, 0), m = 2
11. (0, 4), m = -3
12. (0, 350), m = 1 −
5
Study Guide and Intervention
Writing Equations in Slope-Intercept Form
Write an Equation Given the Slope and a Point
Write an equation of
the line that passes through (-4, 2)
with a slope of 3.
The line has slope 3. To find the
y-intercept, replace m with 3 and (x, y)
with (-4, 2) in the slope-intercept form.
Then solve for b.
y = mx + b
Slope-intercept form
2 = 3(-4) + b
m = 3, y = 2, and x = -4
2 = -12 + b
Multiply.
14 = b
Add 12 to each side.
Therefore, the equation is y = 3x + 14.
Write an equation of the line
that passes through (-2, -1) with a
slope of 1 −
4
.
The line has slope 1 −
4
. Replace m with 1 −
4
and (x, y)
with (-2, -1) in the slope-intercept form.
y = mx + b
Slope-intercept form
-1 = 1 −
4
(-2) + b
m = 1
−
4
, y = -1, and x = -2
-1 = - 1 −
2
+ b
Multiply.
- 1 −
2
= b
Add 1
−
2
to each side.
Therefore, the equation is y = 1 −
4
x - 1 −
2
.
Example 1
Example 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4-2
Chapter 4
12
Glencoe Algebra 1
Study Guide and Intervention (continued)
Writing Equations in Slope-Intercept Form
Write an Equation Given Two Points
Write an equation of the line that passes through (1, 2) and (3, -2).
Find the slope m. To find the y-intercept, replace m with its computed value and (x, y) with
(1, 2) in the slope-intercept form. Then solve for b.
m =
y 2 - y 1 −
x 2 - x 1
Slope formula
m = -2 - 2 −
3 - 1
y2 = -2, y1 = 2, x2 = 3, x1 = 1
m = -2
Simplify.
y = mx + b
Slope-intercept form
2 = -2(1) + b
Replace m with -2, y with 2, and x with 1.
2 = -2 + b
Multiply.
4 = b
Add 2 to each side.
Therefore, the equation is y = -2x + 4.
Exercises
Write an equation of the line that passes through each pair of points.
1.
(1, 1)
(0, –3)
x
y
O
2.
(0, 4)
(4, 0)
x
y
O
3.
(0, 1)
(–3, 0)
x
y
O
4. (-1, 6), (7, -10)
5. (0, 2), (1, 7)
6. (6, -25), (-1, 3)
7. (-2, -1), (2, 11)
8. (10, -1), (4, 2)
9. (-14, -2), (7, 7)
10. (4, 0), (0, 2)
11. (-3, 0), (0, 5)
12. (0, 16), (-10, 0)
Example
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-2
Chapter 4
13
Glencoe Algebra 1
Write an equation of the line that passes through the given point with the
given slope.
1.
(–1, 4)
x
y
O
m = –3
2.
(4, 1)
x
y
O
m = 1
3.
(
-1, 2)
x
y
O
m = 2
4. (1, 9); slope 4
5. (4, 2); slope -2
6. (2, -2); slope 3
7. (3, 0); slope 5
8. (-3, -2); slope 2
9. (-5, 4); slope -4
Write an equation of the line that passes through each pair of points.
10.
(–2, 3)
(3, –2)
x
y
O
11.
(–1, –3)
(1, 1)
x
y
O
12.
(2, –1)
(0, 3)
x
y
O
13. (1, 3), (-3, -5)
14. (1, 4), (6, -1)
15. (1, -1), (3, 5)
16. (-2, 4), (0, 6)
17. (3, 3), (1, -3)
18. (-1, 6), (3, -2)
19. INVESTING The price of a share of stock in XYZ Corporation was $74 two weeks ago.
Seven weeks ago, the price was $59 a share.
a. Write a linear equation to find the price p of a share of XYZ Corporation stock
w weeks from now.
b. Estimate the price of a share of stock five weeks ago.
Skills Practice
Writing Equations in Slope-Intercept Form
4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
14
Glencoe Algebra 1
Practice
Writing Equations in Slope-Intercept Form
Write an equation of the line that passes through the given point and has the
given slope.
1.
(1, 2)
x
y
O
m = 3
2.
(–2, 2)
x
y
O
m = –2
3.
(–1, –3)
x
y
O
m = –1
4. (-5, 4); slope -3
5. (4, 3); slope 1 −
2
6. (1, -5); slope - 3 −
2
7. (3, 7); slope 2 −
7
8.
(
-2, 5 −
2
)
; slope - 1 − 2
9. (5, 0); slope 0
Write an equation of the line that passes through each pair of points.
10.
(4, –2)
(2, –4)
x
y
O
11.
(0, 5)
(4, 1)
x
y
O
12.
(–3, 1)
(–1, –3)
x
y
O
13. (0, -4), (5, -4)
14. (-4, -2), (4, 0)
15. (-2, -3), (4, 5)
16. (0, 1), (5, 3)
17. (-3, 0), (1, -6)
18. (1, 0), (5, -1)
19. DANCE LESSONS The cost for 7 dance lessons is $82. The cost for 11 lessons is $122.
Write a linear equation to find the total cost C for ℓ lessons. Then use the equation to
find the cost of 4 lessons.
20. WEATHER It is 76°F at the 6000-foot level of a mountain, and 49°F at the 12,000-foot
level of the mountain. Write a linear equation to find the temperature T at an elevation
x on the mountain, where x is in thousands of feet.
4-2
Lesson X-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-2
Chapter 4
15
Glencoe Algebra 1
Word Problem Practice
Writing Equations in Slope-Intercept Form
1. FUNDRAISING Yvonne and her friends
held a bake sale to benefit a shelter for
homeless people. The friends sold
22 cakes on the first day and 15 cakes on
the second day of the bake sale. They
collected $88 on the first day and $60 on
the second day. Let x represent the
number of cakes sold and y represent the
amount of money made. Find the slope of
the line that would pass through the
points given.
2. JOBS Mr. Kimball receives a $3000
annual salary increase on the
anniversary of his hiring if he receives
a satisfactory performance review.
His starting salary was $41,250. Write
an equation to show k, Mr. Kimball’s
salary after t years at this company
if his performance reviews are
always satisfactory.
3. CENSUS The population of Laredo,
Texas, was about 215,500 in 2007. It
was about 123,000 in 1990. If we assume
that the population growth is constant
and t represents the number of years
after 1990, write a linear equation to
find p, Laredo’s population for any year
after 1990.
4. WATER Mr. Williams pays $40 a month
for city water, no matter how many
gallons of water he uses in a given
month. Let x represent the number of
gallons of water used per month. Let y
represent the monthly cost of the city
water in dollars. What is the equation of
the line that represents this information?
What is the slope of the line?
5. SHOE SIZES The table shows how
women’s shoe sizes in the United
Kingdom compare to women’s shoe sizes
in the United States.
Women’s Shoe Sizes
U.K.
3
3.5
4
4.5
5
5.5
6
U.S. 5.5
6
6.5
7
7.5
8
8.5
Source: DanceSport UK
a. Write a linear equation to determine
any U.S. size y if you are given the
U.K. size x.
b. What are the slope and y-intercept of
the line?
c. Is the y-intercept a valid data point
for the given information?
4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
16
Glencoe Algebra 1
Tangent to a Curve
A tangent line is a line that intersects a curve at a point with the same rate of change, or
slope, as the rate of change of the curve at that point.
For quadratic functions, functions of the form y = ax2 + bx + c, equations of the tangent lines
can be found. This is based on the fact that the slope through any two points on the curve
is equal to the slope of the line tangent to the curve at the point whose x-value is halfway
between the x-values of the other two points.
Find an equation of the line tangent to the
curve y = x2 + 3x + 2 through the point (2, 12).
First find two points on the curve whose x-values are
equidistant from the x-value of (2, 12).
Step 1: Find two points on the curve. Use x = 1 and x = 3.
When x = 1, y = 12 + 3(1) + 2 or 6.
When x = 3, y = 32 + 3(3) + 2 or 20.
So, the two ordered pairs are (1, 6) and (3, 20).
Step 2: Find the slope of the line that passes through these two points.
m = 20 - 6 −
3 - 1
or 7
Step 3: Now use this slope and the point (2, 12) to find an equation of the tangent line.
y = mx + b
Slope-intercept form
12 = 7(2) + b
Replace x with 2, y with 12, and m with 7.
-2 = b
Solve for b.
So, an equation of the tangent line to y = x2 + 3x + 2 through the point (2, 12) is y = 7x – 2.
Exercises
Find an equation of the line tangent to each curve through the
given point.
1. y = x2 - 3x + 7, (2, 5)
2. y = 3x2 + 4x - 5, (-4, 27)
3. y = 5 - x2, (1, 4)
4. Find the slope of the line tangent to the curve at x = 0 for the general equation
y = ax2 + bx + c.
5. Find the slope of the line tangent to the curve y = ax2 + bx + c at x by finding the slope
of the line through the points (0, c) and (2x, 4ax2 + 2bx + c). Does this equation find the
same slope for x = 0 as you found in Exercise 4?
Enrichment
y
x
O
Example
4-2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-3
Chapter 4
17
Glencoe Algebra 1
Study Guide and Intervention
Writing Equations in Point-Slope Form
Point-Slope Form
Point-Slope Form
y - y1 = m(x - x1), where (x1, y1) is a given point on a nonvertical line
and m is the slope of the line
Write an equation in
point-slope form for the line that passes
through (6, 1) with a slope of -
5
− 2 .
y - y1 = m(x - x1)
Point-slope form
y - 1 = - 5 −
2
(x - 6) m = - 5 −
2
; (x1, y1) = (6, 1)
Therefore, the equation is y - 1 = - 5 −
2
(x - 6).
Write an equation in
point-slope form for a horizontal line
that passes through (4, -1).
y - y1 = m(x - x1)
Point-slope form
y - (-1) = 0(x - 4)
m = 0; (x1, y1) = (4, -1)
y + 1 = 0
Simplify.
Therefore, the equation is y + 1 = 0.
Exercises
Write an equation in point-slope form for the line that passes through each point
with the given slope.
1.
(4, 1)
x
y
O
m = 1
2.
(–3, 2)
x
y
O
m = 0
3.
(2, –3)
x
y
O
m = –2
4. (2, 1), m = 4
5. (-7, 2), m = 6
6. (8, 3), m = 1
7. (-6, 7), m = 0
8. (4, 9), m = 3 −
4
9. (-4, -5), m = - 1 −
2
10. Write an equation in point-slope form for a horizontal line that passes through
(4, -2).
11. Write an equation in point-slope form for a horizontal line that passes through
(-5, 6).
12. Write an equation in point-slope form for a horizontal line that passes through (5, 0).
Example 1
Example 2
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
18
Glencoe Algebra 1
Study Guide and Intervention (continued)
Writing Equations in Point-Slope Form
Forms of Linear Equations
Slope-Intercept
Form
y = mx + b
m = slope; b = y-intercept
Point-Slope
Form
y - y1 = m(x - x1)
m = slope; (x1, y1) is a given point
Standard
Form
Ax + By = C
A and B are not both zero. Usually A is nonnegative and A, B, and
C are integers whose greatest common factor is 1.
Write y + 5 = 2 −
3
(x - 6) in
standard form.
y + 5 = 2 −
3
(x - 6)
Original equation
3(y + 5) = 3 (
2 −
3
) (x - 6) Multiply each side by 3.
3y + 15 = 2(x - 6)
Distributive Property
3y + 15 = 2x - 12
Distributive Property
3y = 2x - 27
Subtract 15 from each side.
-2x + 3y = -27
Add -2x to each side.
2x - 3y = 27
Multiply each side by -1.
Therefore, the standard form of the equation
is 2x - 3y = 27.
Write y - 2 = - 1 −
4
(x - 8) in
slope-intercept form.
y - 2 = - 1 −
4
(x - 8)
Original equation
y - 2 = - 1 −
4
x + 2
Distributive Property
y = - 1 −
4
x + 4
Add 2 to each side.
Therefore, the slope-intercept form of the
equation is y = - 1 −
4
x + 4.
Exercises
Write each equation in standard form.
1. y + 2 = -3(x - 1)
2. y - 1 = - 1 −
3
(x - 6)
3. y + 2 = 2 −
3
(x - 9)
4. y + 3 = -(x - 5)
5. y - 4 = 5 −
3
(x + 3)
6. y + 4 = - 2 −
5
(x - 1)
Write each equation in slope-intercept form.
7. y + 4 = 4(x - 2)
8. y - 5 = 1 −
3
(x - 6)
9. y - 8 = - 1 −
4
(x + 8)
10. y - 6 = 3 (x -
1 −
3
)
11. y + 4 = -2(x + 5)
12. y + 5 −
3
= 1 −
2
(x - 2)
Example 1
Example 2
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-3
Chapter 4
19
Glencoe Algebra 1
Skills Practice
Writing Equations in Point-Slope Form
Write an equation in point-slope form for the line that passes through each point
with the given slope.
1.
(–1, –2)
x
y
O
m = 3
2.
(1, –2)
x
y
O
m = –1
3.
(2, –3)
x
y
O
m = 0
4. (3, 1), m = 0
5. (-4, 6), m = 8
6. (1, -3), m = -4
7. (4, -6), m = 1
8. (3, 3), m = 4 −
3
9. (-5, -1), m = - 5 −
4
Write each equation in standard form.
10. y + 1 = x + 2
11. y + 9 = -3(x - 2)
12. y - 7 = 4(x + 4)
13. y - 4 = -(x - 1)
14. y - 6 = 4(x + 3)
15. y + 5 = -5(x - 3)
16. y - 10 = -2(x - 3)
17. y - 2 = - 1 −
2
(x - 4)
18. y + 11 = 1 −
3
(x + 3)
Write each equation in slope-intercept form.
19. y - 4 = 3(x - 2)
20. y + 2 = -(x + 4)
21. y - 6 = -2(x + 2)
22. y + 1 = -5(x - 3)
23. y - 3 = 6(x - 1)
24. y - 8 = 3(x + 5)
25. y - 2 = 1 −
2
(x + 6)
26. y + 1 = - 1 −
3
(x + 9)
27. y - 1 −
2
= x + 1 −
2
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
20
Glencoe Algebra 1
Practice
Writing Equations in Point-Slope Form
Write an equation in point-slope form for the line that passes through each point
with the given slope.
1. (2, 2), m = -3
2. (1, -6), m = -1
3. (-3, -4), m = 0
4. (1, 3), m = - 3 −
4
5. (-8, 5), m = - 2 −
5
6. (3, -3), m = 1 −
3
Write each equation in standard form.
7. y - 11 = 3(x - 2)
8. y - 10 = -(x - 2)
9. y + 7 = 2(x + 5)
10. y - 5 = 3 −
2
(x + 4)
11. y + 2 = - 3 −
4
(x + 1)
12. y - 6 = 4 −
3
(x - 3)
13. y + 4 = 1.5(x + 2)
14. y - 3 = -2.4(x - 5)
15. y - 4 = 2.5(x + 3)
Write each equation in slope-intercept form.
16. y + 2 = 4(x + 2)
17. y + 1 = -7(x + 1)
18. y - 3 = -5(x + 12)
19. y - 5 = 3 −
2
(x + 4)
20. y - 1 −
4
= - 3 (x +
1 −
4
)
21. y - 2 −
3
= -2 (x -
1 −
4
)
22. CONSTRUCTION A construction company charges $15 per hour for debris removal,
plus a one-time fee for the use of a trash dumpster. The total fee for 9 hours of service
is $195.
a. Write the point-slope form of an equation to find the total fee y for any number of
hours x.
b. Write the equation in slope-intercept form.
c. What is the fee for the use of a trash dumpster?
23. MOVING There is a daily fee for renting a moving truck, plus a charge of $0.50 per mile
driven. It costs $64 to rent the truck on a day when it is driven 48 miles.
a. Write the point-slope form of an equation to find the total charge y for a one-day
rental with x miles driven.
b. Write the equation in slope-intercept form.
c. What is the daily fee?
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-3
Chapter 4
21
Glencoe Algebra 1
1. BICYCLING Harvey rides his bike at an
average speed of 12 miles per hour. In
other words, he rides 12 miles in 1 hour,
24 miles in 2 hours, and so on. Let h be
the number of hours he rides and d be
distance traveled. Write an equation for
the relationship between distance and
time in point-slope form.
2. GEOMETRY The perimeter of a square
varies directly with its side length. The
point-slope form of the equation for this
function is y - 4 = 4(x - 1). Write the
equation in standard form.
3. NATURE The frequency of a male
cricket’s chirp is related to the outdoor
temperature. The relationship is
expressed by the equation T = n + 40,
where T is the temperature in degrees
Fahrenheit and n is the number of chirps
the cricket makes in 14 seconds. Use
the information from the graph below to
write an equation for the line in point-
slope form .
Number of Chirps
15
10
5
0
25
20
y
x
30 35
Temperature (°F)
30
40
20
10
50
70
60
4. CANOEING Geoff paddles his canoe at
an average speed of 3.5 miles per hour.
After 5 hours of canoeing, Geoff has
traveled 18 miles. Write an equation in
point-slope form to find the total distance
y for any number of hours x.
5. AVIATION A jet plane takes off and
consistently climbs 20 feet for every
40 feet it moves horizontally. The graph
shows the trajectory of the jet.
Horizontal Distance (ft)
500
0
1000
1500 2000
2500
Height (ft)
600
800
400
200
1000
1400
1200
a. Write an equation in point-slope form
for the line representing the jet’s
trajectory.
b. Write the equation from part a in
slope -intercept form.
c. Write the equation in standard form.
Word Problem Practice
Writing Equations in Point-Slope Form
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
22
Glencoe Algebra 1
Enrichment
x
y
O
x
y
O
Collinearity
You have learned how to find the slope between two points on a line. Does
it matter which two points you use? How does your choice of points affect
the slope-intercept form of the equation of the line?
1. Choose three different pairs of points from the graph at the
right. Write the slope-intercept form of the line using each pair.
2. How are the equations related?
3. What conclusion can you draw from your answers to Exercises 1 and 2?
When points are contained in the same line, they are said to be collinear.
Even though points may look like they form a line when connected, it does
not mean that they actually do. By checking pairs of points on a graph
you can determine whether the graph represents a linear relationship.
4. Choose several pairs of points from the graph at the right
and write the slope-intercept form of the line containing
each pair.
5. What conclusion can you draw from your equations in
Exercise 4? Is this a line?
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-3
Chapter 4
23
Glencoe Algebra 1
Graphing Calculator Activity
Writing Linear Equations
Lists can be used with the linear regression function to write and verify
linear equations given two points on a line, or the slope of a line and a point
through which it passes. The linear regression function, LinReg (ax + b), is
found under the STAT CALC menu.
Write the slope-intercept form of an equation of the
line that passes through (3, -2) and (6, 4).
Enter the x-coordinates of the points into L1 and the y-coordinates
into L2. Use the linear regression function to write the equation of
the line.
Keystrokes: STAT ENTER 3 ENTER 6 ENTER
(–) 2 ENTER 4
ENTER STAT
4 2nd
[L1]
, 2nd
[L2] ENTER .
The equation is y = 2x - 8.
If you have already written the equation of a line, you can use the given information to
verify your equation.
Exercises
Write the slope-intercept form and the standard form of an equation of the line
that satisfies each condition.
1. passes through (0, 7) and (
1 −
7
, -5)
2. passes through (-5, 1), (10, 10), and (-10, -2)
3. passes through (6, -4), m = 2 −
3
4. passes through (3, 5), m = -4
5. x-intercept: 1, y-intercept: - 1 −
2
6. passes through (-18, 11), y-intercept: 3
Verify that the equation of the line passing through
(2, -3) with slope -
3
− 4 can be written as 3x + 4y = -6.
Use the given point and slope to determine a second point through
which the line passes. Enter the x-coordinates of the points into L1
and the y-coordinates into L2. Use LinReg (ax + b) to determine
the slope-intercept form of the equation.
The slope-intercept form of the equation is y = -0.75x - 1.5 or y = - 3 −
4
x - 3 −
2
.
This can be rewritten in standard form as 3x + 4y = -6.
Example 1
Example 2
4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
24
Glencoe Algebra 1
Study Guide and Intervention
Parallel and Perpendicular Lines
Parallel Lines Two nonvertical lines are parallel if they have the same slope. All
vertical lines are parallel.
Write an equation in slope-intercept form for the line that passes
through (-1, 6) and is parallel to the graph of y = 2x + 12.
A line parallel to y = 2x + 12 has the same slope, 2. Replace m with 2 and (x1, y1) with
(-1, 6) in the point-slope form.
y - y1 = m(x - x1)
Point-slope form
y - 6 = 2(x - (-1)) m = 2; (x1, y1) = (-1, 6)
y - 6 = 2(x + 1)
Simplify.
y - 6 = 2x + 2
Distributive Property
y = 2x + 8
Slope-intercept form
Therefore, the equation is y = 2x + 8.
Exercises
Write an equation in slope-intercept form for the line that passes through the
given point and is parallel to the graph of each equation.
1.
2.
3.
4. (-2, 2), y = 4x - 2
5. (6, 4), y = 1 −
3
x + 1
6. (4, -2), y = -2x + 3
7. (-2, 4), y = -3x + 10
8. (-1, 6), 3x + y = 12
9. (4, -6), x + 2y = 5
10. Find an equation of the line that has a y-intercept of 2 that is parallel to the graph of
the line 4x + 2y = 8.
11. Find an equation of the line that has a y-intercept of -1 that is parallel to the graph of
the line x - 3y = 6.
12. Find an equation of the line that has a y-intercept of -4 that is parallel to the graph of
the line y = 6.
(–3, 3)
x
y
O
4x - 3y = –12
(
-8, 7)
x
y
O
y = - x - 4
1
2
2
2
(5, 1)
x
y
O
y = x - 8
Example
4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-4
Chapter 4
25
Glencoe Algebra 1
Study Guide and Intervention (continued)
Parallel and Perpendicular Lines
Perpendicular Lines Two nonvertical lines are perpendicular if their slopes are
negative reciprocals of each other. Vertical and horizontal lines are perpendicular.
Write an equation in slope-intercept form for the line that passes
through (-4, 2) and is perpendicular to the graph of 2x - 3y = 9.
Find the slope of 2x - 3y = 9.
2x - 3y = 9
Original equation
-3y = -2x + 9 Subtract 2x from each side.
y = 2 −
3
x - 3
Divide each side by -3.
The slope of y = 2 −
3
x - 3 is 2 −
3
. So, the slope of the line passing through (-4, 2) that is
perpendicular to this line is the negative reciprocal of 2 −
3
, or - 3 −
2
.
Use the point-slope form to find the equation.
y - y1 = m(x - x1)
Point-slope form
y - 2 = - 3 −
2
(x - (-4))
m = - 3 −
2
; (x1, y1) = (-4, 2)
y - 2 = - 3 −
2
(x + 4)
Simplify.
y - 2 = - 3 −
2
x - 6
Distributive Property
y = - 3 −
2
x - 4
Slope-intercept form
Exercises
1. ARCHITECTURE On the architect’s plans for a new high school, a wall represented
by
−−−
MN has endpoints M(-3, -1) and N(2, 1). A wall represented by
−−−
PQ has endpoints
P(4, -4) and Q(-2, 11). Are the walls perpendicular? Explain.
Determine whether the graphs of the following equations are parallel or
perpendicular.
2. 2x + y = -7, x - 2y = -4, 4x - y = 5
3. y = 3x, 6x - 2y = 7, 3y = 9x - 1
Write an equation in slope-intercept form for the line that passes through the
given point and is perpendicular to the graph of each equation.
4. (4, 2), y = 1 −
2
x + 1
5. (2, -3), y = - 2 −
3
x + 4
6. (6, 4), y = 7x + 1
7. (-8, -7), y = -x - 8
8. (6, -2), y = -3x - 6
9. (-5, -1), y = 5 −
2
x - 3
Example
4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
26
Glencoe Algebra 1
Skills Practice
Parallel and Perpendicular Lines
Write an equation in slope-intercept form for the line that passes through the
given point and is parallel to the graph of the given equation.
1.
2.
3.
4. (3, 2), y = 3x + 4
5. (-1, -2), y = -3x + 5
6. (-1, 1), y = x - 4
7. (1, -3), y = -4x - 1
8. (-4, 2), y = x + 3
9. (-4, 3), y = 1 −
2
x - 6
10. RADAR On a radar screen, a plane located at A(-2, 4) is flying toward B(4, 3).
Another plane, located at C(-3, 1), is flying toward D(3, 0). Are the planes’ paths
perpendicular? Explain.
Determine whether the graphs of the following equations are parallel or
perpendicular. Explain.
11. y = 2 −
3
x + 3, y = 3 −
2
x, 2x - 3y = 8
12. y = 4x, x + 4 y = 12, 4x + y = 1
Write an equation in slope-intercept form for the line that passes through the
given point and is perpendicular to the graph of the given equation.
13. (-3, -2), y = x + 2
14. (4, -1), y = 2x - 4
15. (-1, -6), x + 3y = 6
16. (-4, 5), y = -4x - 1
17. (-2, 3), y = 1 −
4
x - 4
18. (0, 0), y = 1 −
2
x - 1
(–2, 2)
x
y
O
y = 12 x + 1
(1, –1)
x
y
O
y = –x + 3
(–2, –3)
x
y
O
y = 2x - 1
4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-4
Chapter 4
27
Glencoe Algebra 1
Practice
Parallel and Perpendicular Lines
Write an equation in slope-intercept form for the line that passes through the
given point and is parallel to the graph of the given equation.
1. (3, 2), y = x + 5
2. (-2, 5), y = -4x + 2
3. (4, -6), y = - 3 −
4
x + 1
4. (5, 4), y = 2 −
5
x - 2
5. (12, 3), y = 4 −
3
x + 5
6. (3, 1), 2x + y = 5
7. (-3, 4), 3y = 2x - 3
8. (-1, -2), 3x - y = 5
9. (-8, 2), 5x - 4y = 1
10. (-1, -4), 9x + 3y = 8
11. (-5, 6), 4x + 3y = 1
12. (3, 1), 2x + 5y = 7
Write an equation in slope-intercept form for the line that passes through the
given point and is perpendicular to the graph of the given equation.
13. (-2, -2), y = - 1 −
3
x + 9
14. (-6, 5), x - y = 5
15. (-4, -3), 4x + y = 7
16. (0, 1), x + 5y = 15
17. (2, 4), x - 6y = 2
18. (-1, -7), 3x + 12y = -6
19. (-4, 1), 4x + 7y = 6
20. (10, 5), 5x + 4y = 8
21. (4, -5), 2x - 5y = -10
22. (1, 1), 3x + 2y = -7
23. (-6, -5), 4x + 3y = -6
24. (-3, 5), 5x - 6y = 9
25. GEOMETRY Quadrilateral ABCD has diagonals −−
AC and
−−−
BD .
Determine whether
−−
AC is perpendicular to
−−−
BD . Explain.
26. GEOMETRY Triangle ABC has vertices A(0, 4), B(1, 2), and C(4, 6). Determine whether
triangle ABC is a right triangle. Explain.
x
y
O
A
D
C
B
4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
28
Glencoe Algebra 1
1. BUSINESS Brady’s Books is a retail
store. The store’s average daily profits y
are given by the equation y = 2x + 3
where x is the number of hours available
for customer purchases. Brady’s adds an
online shopping option. Write an
equation in slope-intercept form to show
a new profit line with the same profit
rate containing the point (0, 12).
2. ARCHITECTURE The front view of a
house is drawn on graph paper. The left
side of the roof of the house is
represented by the equation y = x. The
rooflines intersect at a right angle and
the peak of the roof is represented by the
point (5, 5). Write the equation in slope-
intercept form for the line that creates
the right side of the roof.
3. ARCHAEOLOGY An archaeologist is
comparing the location of a jeweled box
she just found to the location of a brick
wall. The wall can be represented by the
equation y = - 5 −
3
x + 13. The box is
located at the point (10, 9). Write an
equation representing a line that is
perpendicular to the wall and that passes
through the location of the box.
4. GEOMETRY A parallelogram is created
by the intersections of the lines x = 2,
x = 6, y = 1 −
2
x + 2, and another line. Find
the equation of the fourth line needed to
complete the parallelogram. The line
should pass through (2, 0). (Hint: Sketch
a graph to help you see the lines.)
5. INTERIOR DESIGN Pamela is planning
to install an island in her kitchen. She
draws the shape she likes by connecting
the vertices of the square tiles on her
kitchen floor. She records the location of
each corner in the table.
a. How many pairs of parallel sides are
there in the shape ABCD she
designed? Explain.
b. How many pairs of perpendicular
sides are there in the shape she
designed? Explain.
c. What is the shape of her new island?
Word Problem Practice
Parallel and Perpendicular Lines
y
x
O
(5, 5)
Corner
Distance
from West
Wall (tiles)
Distance
from South
Wall (tiles)
A
5
4
B
3
8
C
7
10
D
11
7
4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-4
Chapter 4
29
Glencoe Algebra 1
Enrichment
Pencils of Lines
All of the lines that pass through
a single point in the same plane
are called a pencil of lines.
All lines with the same slope,
but different intercepts, are also
called a “pencil,” a pencil of
parallel lines.
Graph some of the lines in each pencil.
1. A pencil of lines through the
2. A pencil of lines described by
point (1, 3)
y - 4 = m(x - 2), where m is any
real number
3. A pencil of lines parallel to the line
4. A pencil of lines described by
x - 2y = 7
y = mx + 3m - 2 , where m is any
real number
x
y
O
x
y
O
x
y
O
x
y
O
4-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
30
Glencoe Algebra 1
Study Guide and Intervention
Scatter Plots and Lines of Fit
Investigate Relationships Using Scatter Plots A scatter plot is a graph in
which two sets of data are plotted as ordered pairs in a coordinate plane. If y increases as x
increases, there is a positive correlation between x and y. If y decreases as x increases,
there is a negative correlation between x and y. If x and y are not related, there is no
correlation.
EARNINGS The graph at the right
shows the amount of money Carmen earned each
week and the amount she deposited in her savings
account that same week. Determine whether the
graph shows a positive correlation, a negative
correlation, or no correlation. If there is a
positive or negative correlation, describe its
meaning in the situation.
The graph shows a positive correlation. The more
Carmen earns, the more she saves.
Exercises
Determine whether each graph shows a positive correlation, a negative
correlation, or no correlation. If there is a positive or negative correlation,
describe its meaning in the situation.
1.
2.
3.
4.
Average Weekly Work Hours in U.S.
Hours
34.0
34.2
33.8
33.6
34.4
34.6
Years Since 1995
3
2
1
0
5
4
7
6
9
8
Source: The World Almanac
Average Jogging Speed
Minutes
Miles per Hour
0
10
20
5
15
25
10
5
Carmen’s Earnings and Savings
Dollars Earned
Dollars Saved
0
40
80
120
35
30
25
20
15
10
5
Example
Average U.S. Hourly
Earnings
Hourly Earnings ($)
15
0
16
17
18
19
Years Since 2003
Source: U.S. Dept. of Labor
1
2
3
4
5
U.S. Imports from Mexico
Imports ($ billions)
130
0
160
190
220
Years Since 2003
Source: U.S. Census Bureau
1
2
3
4
5
4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-5
Chapter 4
31
Glencoe Algebra 1
Use Lines of Fit
The table shows the number of students per computer in Easton
High School for certain school years from 1996 to 2008.
Year
1996
1998
2000
2002
2004
2006
2008
Students per Computer
22
18
14
10
6.1
5.4
4.9
a. Draw a scatter plot and determine
what relationship exists, if any.
Since y decreases as x increases, the
correlation is negative.
b. Draw a line of fit for the scatter plot.
Draw a line that passes close to most of the points.
A line of fit is shown.
c. Write the slope-intercept form of an equation
for the line of fit.
The line of fit shown passes through
(1999, 16) and (2005, 5.7). Find the slope.
m = 5.7 - 16
−
2005 - 1999
m = -1.7
Find b in y = -1.7x + b.
16 = -1.7 · 1993 + b
3404 = b
Therefore, an equation of a line of fit is y = -1.7x + 3404.
Exercises
Refer to the table for Exercises 1–3.
1. Draw a scatter plot.
2. Draw a line of fit for the data.
3. Write the slope-intercept
form of an equation for the
line of fit.
Movie Admission Prices
Admission ($)
5.4
5.6
5.2
5
5.8
6
6.2
Years Since 1999
3
2
1
0
5
4
Source: U.S. Census Bureau
Study Guide and Intervention (continued)
Scatter Plots and Lines of Fit
Students per Computer
in Easton High School
Students per Computer
8
12
16
4
0
20
24
Year
1996 1998 2000 2002 2004 2006 2008
Example
Years
Since 1999
Admission
(dollars)
0
$5.08
1
$5.39
2
$5.66
3
$5.81
4
$6.03
4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
32
Glencoe Algebra 1
Skills Practice
Scatter Plots and Lines of Fit
Determine whether each graph shows a positive correlation, a negative
correlation, or no correlation. If there is a positive or negative correlation,
describe its meaning in the situation.
1.
2.
3.
4.
5. BASEBALL The scatter plot shows the average price of a major-league baseball ticket
from 1997 to 2006.
a. Determine what relationship, if any, exists in the
data. Explain.
b. Use the points (1998, 13.60) and (2003, 19.00)
to write the slope-intercept form of an equation for
the line of fit shown in the scatter plot.
c. Predict the price of a ticket in 2009.
Weight-Lifting
Weight (pounds)
Repetitions
0
40
80
20
60
100 120 140
14
12
10
8
6
4
2
Library Fines
Books Borrowed
Fines (dollars)
0
2
4
5
6
7
8
9
1
3
10
7
6
5
4
3
2
1
Calories Burned
During Exercise
Time (minutes)
Calories
0
20
40
10
30
50 60
600
500
400
300
200
100
Baseball Ticket Prices
Average Price ($)
14
16
12
0
18
20
22
24
Year
’99
’98
’97
’01
’03
’00
Source: Team Marketing Report, Chicago
’02
’04 ’05 ’06
Car Dealership Revenue
Revenue
(hundreds of thousands)
4
6
2
0
8
10
12
14
Year
’99
’01
’03
’00
’02
’04 ’05 ’06 ’07 ’08
4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-5
Chapter 4
33
Glencoe Algebra 1
Practice
Scatter Plots and Lines of Fit
Determine whether each graph shows a positive correlation, a negative
correlation, or no correlation. If there is a positive or negative correlation,
describe its meaning in the situation.
1.
2.
3. DISEASE The table shows the number of cases of
Foodborne Botulism in the United States for the
years 2001 to 2005.
a. Draw a scatter plot and determine what
relationship, if any, exists in the data.
b. Draw a line of fit for the scatter plot.
c. Write the slope-intercept form of an equation for the
line of fit.
4. ZOOS The table shows the average and maximum
longevity of various animals in captivity.
a. Draw a scatter plot and determine what
relationship, if any, exists in the data.
b. Draw a line of fit for the scatter plot.
c. Write the slope-intercept form of an equation for the
line of fit.
d. Predict the maximum longevity for an animal with
an average longevity of 33 years.
State Elevations
Mean Elevation (feet)
Highest Point
(thousands of feet)
1000
0
2000
3000
16
12
8
4
Source: U.S. Geological Survey
Temperature versus Rainfall
Average Annual Rainfall (inches)
Average
Temperature (ºF)
10 15 20 25 30 35 40 45
64
60
56
52
0
Source: National Oceanic and Atmospheric
Administration
U.S. Foodborne
Botulism Cases
Cases
20
30
10
0
40
50
Year
2001
2002
2003
2004
2005
Animal Longevity (Years)
Average
Maximum
5
0
10 15 20 25 30 35 40 45
80
70
60
50
40
30
20
10
Source: Centers for Disease Control
U.S. Foodborne Botulism Cases
Year
2001 2002 2003 2004 2005
Cases
39
28
20
16
18
Source: Walker’s Mammals of the World
Longevity (years)
Avg. 12 25 15
8 35 40 41 20
Max. 47 50 40 20 70 77 61 54
4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
34
Glencoe Algebra 1
1. MUSIC The scatter plot shows the
number of CDs in millions that were sold
from 1999 to 2005. If the trend
continued, about how many CDs were
sold in 2006?
2. FAMILY The table shows the predicted
annual cost for a middle income family to
raise a child from birth until adulthood.
Draw a scatter plot and describe what
relationship exists within the data.
3. HOUSING The median price of an
existing home was $160,000 in 2000 and
$240,000 in 2007. If x represents the
number of years since 2000, use these
data points to determine a line of best fit
for the trends in the price of existing
homes. Write the equation in slope-
intercept form.
4. BASEBALL The table shows the average
length in minutes of professional
baseball games in selected years.
Source: Elias Sports Bureau
a. Draw a scatter plot and determine
what relationship, if any, exists in the
data.
b. Explain what the scatter plot shows.
c. Draw a line of fit for the scatter plot.
Time (min)
166
0
168
170
172
174
176
178
180
Year
’90 ’92 ’94 ’96 ’98 ’00 ’02
Age (years)
3
0
6
12
15
y
x
9
Annual Cost ($1000)
11
12
10
9
13
16
15
14
17
Source: The World Almanac
Source: RIAA
Year
‘01
‘00
‘99
0
‘03
‘02
‘05
y
x
‘04
CDs (millions)
750
800
700
650
850
950
900
Word Problem Practice
Scatter Plots and Lines of Fit
Cost of Raising a Child Born in 2003
Child’s
Age
3
6
9
12
15
Annual
Cost ($)
10,700 11,700 12,600 15,000 16,700
Average Length of
Major League Baseball Games
Year
‘92
‘94
‘96
‘98
‘00
‘02
‘04
Time (min) 170 174 171 168 178 172 167
4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-5
Chapter 4
35
Glencoe Algebra 1
Enrichment
Latitude and Temperature
The latitude of a place on Earth
is the measure of its distance from
the equator. What do you think is
the relationship between a city’s
latitude and its mean January
temperature? At the right is a
table containing the latitudes and
January mean temperatures for
fifteen U.S. cities.
Sources: National Weather Service
1. Use the information in the table to create
a scatter plot and draw a line of best fit
for the data.
2. Write an equation for the line of fit. Make
a conjecture about the relationship
between a city’s latitude and its mean
January temperature.
3. Use your equation to predict the January
mean temperature of Juneau, Alaska, which has latitude 58:23 N.
4. What would you expect to be the latitude of a city with a January mean temperature
of 15°F?
5. Was your conjecture about the relationship between latitude and temperature correct?
6. Research the latitudes and temperatures for cities in the southern hemisphere. Does
your conjecture hold for these cities as well?
Latitude (ºN)
Temperature (
º
F)
70
60
50
40
30
20
10
0
-10
T
L
20
40
60
10
30
50
U.S. City
Latitude January Mean Temperature
Albany, New York
42:40 N
20.7°F
Albuquerque, New Mexico
35:07 N
34.3°F
Anchorage, Alaska
61:11 N
14.9°F
Birmingham, Alabama
33:32 N
41.7°F
Charleston, South Carolina 32:47 N
47.1°F
Chicago, Illinois
41:50 N
21.0°F
Columbus, Ohio
39:59 N
26.3°F
Duluth, Minnesota
46:47 N
7.0°F
Fairbanks, Alaska
64:50 N
-10.1°F
Galveston, Texas
29:14 N
52.9°F
Honolulu, Hawaii
21:19 N
72.9°F
Las Vegas, Nevada
36:12 N
45.1°F
Miami, Florida
25:47 N
67.3°F
Richmond, Virginia
37:32 N
35.8°F
Tucson, Arizona
32:12 N
51.3°F
4-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
36
Glencoe Algebra 1
Exercises
The table shows the number of millions of dollars of direct
political contributions received by Democrats and Republicans
in selected years.
1. Use a spreadsheet to draw a scatter plot and a trendline for the
data. Let x represent the number of years since 1990 and let
y represent direct political contributions in millions of dollars.
2. Predict the amount of direct political contributions for the 2010 election.
Spreadsheet Activity
Scatter Plots
The table below shows the number of metric tons of gold produced
in mines in the United States in selected years.
Use a spreadsheet to draw a scatter plot and a trendline for the data.
Let x represent the number of years since 2000 and let y represent the
number of metric tons of gold. Then predict the number of ounces of
gold produced in 2013.
Step 1 Use Column A for the years since 2000 and Column B for the number of metric
tons of gold. To create a graph from the data, select the data in Columns A and B
and choose Chart from the Insert menu. Select an XY (Scatter) chart to show the
data points.
Step 2 Add a trendline to the graph by choosing the Chart menu. Add a linear
trendline. Use the options menu to have the trendline forecast 5 years forward.
Using this trendline, it appears that the gold production for 2013 will be
approximately 150 metric tons.
A spreadsheet program can create scatter plots of data that you enter. You can also
have the spreadsheet graph a line of fit, called a trendline, automatically.
Example
Source: Open Secrets
Year
Contributions
1990
281
1994
337
1998
445
2002
717
2006
1085
Source: U.S. Geological Survey
Year
2000 2001
2002
2003
2004 2005
2006
2007
2008
2009
Gold
353
335
298
277
247
256
252
238
233
210
4-5
A
1
0
1
2
3
4
5
6
7
8
9
353
335
298
277
247
256
252
238
233
210
3
4
5
6
7
8
9
10
11
12
13
14
2
B
C
D
E
F
G
H
15
Spreadsheet sample
Sheet 1
Sheet 2
Sheet 3
U.S. Gold Mine Production
Gold (metric tons)
100
150
50
0
200
250
300
350
400
Years since 2000
5
10
15
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-6
Chapter 4
37
Glencoe Algebra 1
Equations of Best-Fit Lines Many graphing calculators utilize an algorithm called
linear regression to find a precise line of fit called the best-fit line. The calculator
computes the data, writes an equation, and gives you the correlation coefficent, a
measure of how closely the equation models the data.
GAS PRICES The table shows the price of a gallon of regular
gasoline at a station in Los Angeles, California on January 1 of various years.
Year
2005
2006
2007
2008
2009
2010
Average Price
$1.47
$1.82
$2.15
$2.49
$2.83
$3.04
Source: U.S. Department of Energy
a. Use a graphing calculator to write an equation for the
best-fit line for that data. Enter the data by pressing STAT
and selecting the Edit option. Let the year 2005 be represented
by 0. Enter the years since 2005 into List 1 (L1). Enter the
average price into List 2 (L2).
Then, perform the linear regression by pressing STAT and
selecting the CALC option. Scroll down to LinReg (ax+b) and
press ENTER . The best-fit equation for the regression is shown
to be y = 0.321x + 1.499.
b. Name the correlation coefficient. The correlation coefficient
is the value shown for r on the calculator screen. The correlation
coefficient is about 0.998.
Exercises
Write an equation of the regression line for the data in each table below. Then
find the correlation coefficient.
1. OLYMPICS Below is a table showing the number of gold medals won by the United
States at the Winter Olympics during various years.
Year
1992
1994
1998
2002
2006
2010
Gold Medals
5
6
6
10
9
9
Source: International Olympic Committee
2. INTEREST RATES Below is a table showing the U.S. Federal Reserve’s prime interest
rate on January 1 of various years.
Year
2006
2007
2008
2009
2010
Prime Rate (percent)
7.25
8.25
7.25
3.25
3.25
Source: Federal Reserve Board
Study Guide and Intervention
Regression and Median-Fit Lines
Example
4-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
38
Glencoe Algebra 1
Equations of Median-Fit Lines A graphing calculator can also find another type of
best-fit line called the median-fit line, which is found using the medians of the coordinates
of the data points.
ELECTIONS The table shows the total number of people in millions
who voted in the U.S. Presidential election in the given years.
Year
1980
1984
1988
1992
1996
2004
2008
Voters
86.5
92.7
91.6
104.4
96.3
122.3
131.3
Source: George Mason University
a. Find an equation for the median-fit line. Enter the data by
pressing STAT and selecting the Edit option. Let the year 1980
be represented by 0. Enter the years since 1980 into List 1 (L1).
Enter the number of voters into List 2 (L2).
Then, press STAT and select the CALC option. Scroll down to
Med-Med option and press ENTER . The value of a is the slope,
and the value of b is the y-intercept.
The equation for the median-fit line is y = 1.55x + 83.57.
b. Estimate the number of people who voted in the 2000 U.S.
Presidential election. Graph the best-fit line. Then use the
TRACE feature and the arrow keys until you find a point
where x = 20.
When x = 20, y ≈ 115. Therefore, about 115 million people voted in the 2000 U.S.
Presidential election.
Exercises
Write an equation of the regression line for the data in each table below. Then
find the correlation coefficient.
1. POPULATION GROWTH Below is a table showing the estimated population of Arizona
in millions on July 1st of various years.
Year
2001
2002
2003
2004
2005
2006
Population
5.30
5.44
5.58
5.74
5.94
6.17
Source: U.S. Census Bureau
a. Find an equation for the median-fit line.
b. Predict the population of Arizona in 2009.
2. ENROLLMENT Below is a table showing the number of students enrolled at Happy
Days Preschool in the given years.
Year
2002
2004
2006
2008
2010
Students
130
168
184
201
234
a. Find an equation for the median-fit line.
b. Estimate how many students were enrolled in 2007.
Study Guide and Intervention (continued)
Regression and Median-Fit Lines
Example
4-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-6
Chapter 4
39
Glencoe Algebra 1
Write an equation of the regression line for the data in each table below. Then
find the correlation coefficient.
1. SOCCER The table shows the number of goals a soccer team scored each season
since 2005.
Year
2005
2006
2007
2008
2009
2010
Goals Scored
42
48
46
50
52
48
2. PHYSICAL FITNESS The table shows the percentage of seventh grade students
in public school who met all six of California’s physical fitness standards each year
since 2002.
Year
2002
2003
2004
2005
2006
Percentage
24.0%
36.4%
38.0%
40.8%
37.5%
Source: California Department of Education
3. TAXES The table shows the estimated sales tax revenues, in billions of dollars, for
Massachusetts each year since 2004.
Year
2004
2005
2006
2007
2008
Tax Revenue
3.75
3.89
4.00
4.17
4.47
Source: Beacon Hill Institute
4. PURCHASING The SureSave supermarket chain closely monitors how many diapers are
sold each year so that they can reasonably predict how many diapers will be sold in the
following year.
Year
2006
2007
2008
2009
2010
Diapers Sold
60,200
65,000
66,300
65,200
70,600
a. Find an equation for the median-fit line.
b. How many diapers should SureSave anticipate selling in 2011?
5. FARMING Some crops, such as barley, are very sensitive to how acidic the soil is. To
determine the ideal level of acidity, a farmer measured how many bushels of barley he
harvests in different fields with varying acidity levels.
Soil Acidity (pH)
5.7
6.2
6.6
6.8
7.1
Bushels Harvested
3
20
48
61
73
a. Find an equation for the regression line.
b. According to the equation, how many bushels would the farmer harvest if the soil had
a pH of 10?
c. Is this a reasonable prediction? Explain.
Skills Practice
Regression and Median-Fit Lines
4-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
40
Glencoe Algebra 1
Write an equation of the regression line for the data in each table below. Then
find the correlation coefficient.
1. TURTLES The table shows the number of turtles hatched at a zoo each year since 2006.
Year
2006
2007
2008
2009
2010
Turtles Hatched
21
17
16
16
14
2. SCHOOL LUNCHES The table shows the percentage of students receiving free or
reduced price school lunches at a certain school each year since 2006.
Year
2006
2007
2008
2009
2010
Percentage
14.4%
15.8%
18.3%
18.6%
20.9%
Source: KidsData
3. SPORTS Below is a table showing the number of students signed up to play lacrosse
after school in each age group.
Age
13
14
15
16
17
Lacrosse Players
17
14
6
9
12
4. LANGUAGE The State of California keeps track of how many millions of students are
learning English as a second language each year.
Year
2003
2004
2005
2006
2007
English Learners
1.600
1.599
1.592
1.570
1.569
Source: California Department of Education
a. Find an equation for the median-fit line.
b. Predict the number of students who were learning English in California in 2001.
c. Predict the number of students who were learning English in California in 2010.
5. POPULATION Detroit, Michigan, like a number of large cities, is losing population
every year. Below is a table showing the population of Detroit each decade.
Year
1960
1970
1980
1990
2000
Population (millions)
1.67
1.51
1.20
1.03
0.95
Source: U.S. Census Bureau
a. Find an equation for the regression line.
b. Find the correlation coefficient and explain the meaning of its sign.
c. Estimate the population of Detroit in 2008.
Practice
Regression and Median-Fit Lines
4-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-6
4-6
Chapter 4
41
Glencoe Algebra 1
Word Problem Practice
Regression and Median-Fit Lines
1. FOOTBALL Rutgers University running
back Ray Rice ran for 1732 total yards in
the 2007 regular season. The table below
shows his cumulative total number of
yards ran after select games.
Game
Number
1
3
6
9
12
Cumulative
Yards
184
431
818
1257 1732
Source: Rutgers University Athletics
Use a calculator to find an equation for
the regression line showing the total
yards y scored after x games. What is
the real-world meaning of the value
returned for a?
2. GOLD Ounces of gold are traded by
large investment banks in commodity
exchanges much the same way that
shares of stock are traded. The table
below shows the cost of a single ounce of
gold on the last day of trading in given
years.
Year
2002
2003
2004
2005
2006
Price
$346.70 $414.80
$438.10
$517.20
$636.30
Source: Global Financial Data
Use a calculator to find an equation for
the regression line. Then predict the
price of an ounce of gold on the last day
of trading in 2009. Is this a reasonable
prediction? Explain.
3. GOLF SCORES Emmanuel is practicing
golf as part of his school’s golf team.
Each week he plays a full round of golf
and records his total score. His scorecard
after five weeks is below.
Week
1
2
3
4
5
Golf Score
112
107
108
104
98
Use a calculator to find an equation for
the median-fit line. Then estimate how
many games Emmanuel will have to play
to get a score of 86.
4. STUDENT ELECTIONS The vote totals
for five of the candidates participating in
Montvale High School’s student council
elections and the number of hours each
candidate spent campaigning are shown
in the table below.
Hours
Campaigning
1
3
4
6
8
Votes Received
9
22
24
46
78
a. Use a calculator to find an equation
for the median-fit line.
b. Plot the data points and draw the
median-fit line on the graph below.
Votes Received
20
30
10
0
40
50
60
70
80
Campaign Time (h)
3
2
1
5
7
4
6
8 x
y
c. Suppose a sixth candidate spends
7 hours campaigning. Estimate how
many votes that candidate could
expect to receive.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
42
Glencoe Algebra 1
For some sets of data, a linear equation in the form y = ax + b does not adequately describe
the relationship between data points. The “QuadReg” function on a graphing calculator
will output an equation in the form y = ax2 + bx + c. The value of R2, the coefficient of
determination tells you how closely the parabola fits the data.
The table shows the population of Atlanta in various years.
Year
1970
1980
1990
2000
2005
2007
Population
497,000
425,000
394,017
416,474
470,688
498,109
Source: U.S. Census Bureau
a. Find the equation of a quadratic-regression parabola for the data.
Running a linear regression on the data provides an r value of 0.03, which indicates a
poor fit. The data appears to be a good candidate for a quadratic regression.
Step 1 Enter the data by pressing STAT and selecting the Edit
option. Enter the years since 1970 as your x-values (L1)
and enter the population figures as your y-values (L2).
Step 2 Perform the quadratic regression by pressing STAT and
selecting the CALC option. Scroll down to QuadReg and
press ENTER .
Step 3 Write the equation of the best-fit parabola by rounding the
a, b, and c values on the screen.
The equation for the best-fit parabola is
y = 302.8x2 – 11,480x + 501,227.
b. Find the coefficient of determination.
The coefficient of determination for the parabola is R2 = 0.969,
which indicates a good fit.
c. Use the quadratic-regression parabola to predict the population in 2010.
Graph the best-fit parabola. Then use the TRACE
feature and
the arrow keys until you find a point where x = 40.
When x ≈ 40, y ≈ 525,000. The estimated population will be 525,000.
Exercises
1. The table below shows the average high temperature in Crystal
River, Florida in various months.
Month
Jan (1)
Mar (3)
May (5)
Jul (7)
Sep (9)
Nov (11)
Avg. High (°F)
68°
76°
87°
91°
88°
76°
Source: Country Studies
a. Find the equation of the best-fit parabola.
b. Find the coefficient of determination.
c. Use the quadratic-regression parabola to predict the average high temperature in
April (4th month).
Enrichment
Quadratic Regression Parabolas
Example
4-6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-7
Chapter 4
43
Glencoe Algebra 1
Inverse Relations An inverse relation is the set of ordered pairs obtained by
exchanging the x-coordinates with the y-coordinates of each ordered pair. The domain of a
relation becomes the range of its inverse, and the range of the relation becomes the domain
of its inverse.
Find and graph the inverse of the relation represented by line a.
The graph of the relation passes through (–2, –10), (–1, –7), (0, –4), (1, –1), (2, 2), (3, 5),
and (4, 8).
To find the inverse, exchange the coordinates
of the ordered pairs.
The graph of the inverse passes through the points
(–10, –2), (–7, –1), (–4, 0), (–1, 1), (2, 2), (5, 3), and (8, 4).
Graph these points and then draw the line that passes
through them.
Exercises
Find the inverse of each relation.
1. {(4, 7), (6, 2), (9, –1), (11, 3)}
2. {(–5, –9), (–4, –6), (–2, –4), (0, –3)}
3.
x
y
–8 –15
–2 –11
1
–8
5
1
11
8
4.
x
y
–8
3
–2
9
2
13
6
18
8
19
5.
x
y
–6
14
–5
11
–4
8
–3
5
–2
2
Graph the inverse of each relation.
6.
y
x
O
8
4
−4
−8
4
8
−4
−8
7.
y
x
O
8
4
−4
−8
4
8
−4
−8
8.
y
x
O
8
4
−4
−8
4
8
−4
−8
Study Guide
Inverse Linear Functions
Example
y
x
O
8
4
−4
−8
4
8
−4
−8
(−10, −2)
(−4, 0)
(2, 2)
(8, 4)
a
4-7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
44
Glencoe Algebra 1
Study Guide (continued)
Inverse Linear Functions
Inverse Functions A linear relation that is described by a function has an inverse
function that can generate ordered pairs of the inverse relation. The inverse of the linear
function f (x) can be written as f -1 (x) and is read f of x inverse or the inverse of f of x.
Find the inverse of f (x) = 3 −
4
x + 6.
Step 1 f (x) = 3 −
4
x + 6
Original equation
y = 3 −
4
x + 6
Replace f (x) with y.
Step 2 x = 3 −
4
y + 6
Interchange y and x.
Step 3 x - 6 = 3 −
4
y
Subtract 6 from each side.
4 −
3
(x - 6) = y
Multiply each side by 4 −
3
.
Step 4 4 −
3
(x - 6) = f -1 (x)
Replace y with f -1 (x).
The inverse of f (x) = 3 −
4
x + 6 is f -1 (x) = 4 −
3
(x - 6) or f -1 (x) = 4 −
3
x - 8.
Exercises
Find the inverse of each function.
1. f (x) = 4x - 3
2. f (x) = -3x + 7
3. f (x) = 3 −
2
x - 8
4. f (x) = 16 - 1 −
3
x
5. f (x) = 3(x - 5)
6. f (x) = -15 - 2 −
5
x
7. TOOLS Jimmy rents a chainsaw from the department store to work on his yard.
The total cost C(x) in dollars is given by C(x) = 9.99 + 3.00x, where x is the
number of days he rents the chainsaw.
a. Find the inverse function C -1 (x).
b. What do x and C -1 (x) represent in the context of the inverse function?
c. How many days did Jimmy rent the chainsaw if the total cost
was $27.99?
Example
4-7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-7
Chapter 4
45
Glencoe Algebra 1
Find the inverse of each relation.
1. x
y
–9
–1
–7
–4
–5
–7
–3 –10
–1 –13
2.
x
y
1
8
2
6
3
4
4
2
5
0
3.
x
y
–4
–2
–2
–1
0
1
2
0
4
2
4. {(-3, 2), (-1, 8), (1, 14), (3, 20)}
5. {(5, -3), (2, -9), (-1, -15), (-4, -21)}
6. {(4, 6), (3, 1), (2, -4), (1, -9)}
7. {(-1, 16), (-2, 12), (-3, 8), (-4, 4)}
Graph the inverse of each function.
8.
y
x
O
8
4
−4
−8
4
8
−4
−8
9.
y
x
O
8
4
−4
−8
4
8
−4
−8
10.
y
x
O
8
4
−4
−8
4
8
−4
−8
Find the inverse of each function.
11. f (x) = 8x - 5
12. f (x) = 6(x + 7)
13. f (x) = 3 −
4
x + 9
14. f (x) = -16 + 2 −
5
x
15. f (x) = 3x + 5 −
4
16. f (x) = -4x + 1 −
5
17. LEMONADE Chrissy spent $5.00 on supplies and lemonade powder for her lemonade
stand. She charges $0.50 per glass.
a. Write a function P(x) to represent her profit per glass sold.
b. Find the inverse function, P -1 (x).
c. What do x and P -1 (x) represent in the context of the inverse function?
d. How many glasses must Chrissy sell in order to make a $3 profit?
Skills Practice
Inverse Linear Functions
4-7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
46
Glencoe Algebra 1
Find the inverse of each relation.
1. {(-2, 1), (-5, 0), (-8, -1), (-11, 2)}
2. {(3, 5), (4, 8), (5, 11), (6, 14)}
3. {(5, 11), (1, 6), (-3, 1), (-7, -4)}
4. {(0, 3), (2, 3), (4, 3), (6, 3)}
Graph the inverse of each function.
5.
y
x
O
8
4
−4
−8
4
8
−4
−8
6.
y
x
O
8
4
−4
−8
4
8
−4
−8
7.
y
x
O
8
4
−4
−8
4
8
−4
−8
Find the inverse of each function.
8. f (x) = 6 −
5
x - 3
9. f (x) = 4x + 2 −
3
10. f (x) = 3x - 1 −
6
11. f (x) = 3(3x + 4)
12. f (x) = -5(-x - 6)
13. f (x) = 2x - 3 −
7
Write the inverse of each equation in f -1 (x) notation.
13. 4x + 6y = 24
14. -3y + 5x = 18
15. x + 5y = 12
16. 5x + 8y = 40
17. -4y - 3x = 15 + 2y
18. 2x - 3 = 4x + 5y
19. CHARITY Jenny is running in a charity event. One donor is paying an initial amount of
$20.00 plus an extra $5.00 for every mile that Jenny runs.
a. Write a function D(x) for the total donation for x miles run.
b. Find the inverse function, D -1 (x).
c. What do x and D -1 (x) represent in the context of the inverse function?
Practice
Inverse Linear Functions
4-7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 4-7
Chapter 4
47
Glencoe Algebra 1
Word Problem Practice
Inverse Linear Functions
1. BUSINESS Alisha started a baking
business. She spent $36 initially on
supplies and can make 5 dozen brownies
at a cost of $12. She charges her
customers $10 per dozen brownies.
a. Write a function C(x) to represent
Alisha’s total cost per dozen
brownies.
b. Write a function E(x) to represent
Alisha’s earnings per dozen brownies
sold.
c. Find P (x) = E(x) - C(x). What does
P (x) represent?
d. Find C -1 (x), E -1 (x), and P -1 (x).
e. How many dozen brownies does
Alisha need to sell in order to make
a profit?
2. GEOMETRY The area of the base of a
cylindrical water tank is 66 square feet.
The volume of water in the tank is
dependent on the height of the water h
and is represented by the function
V(h) = 66h. Find V -1 (h). What will the
height of the water be when the volume
reaches 2310 cubic feet?
3. SERVICE A technician is working on a
furnace. He is paid $150 per visit plus
$70 for every hour he works on the
furnace.
a. Write a function C(x) to represent the
total charge for every hour of
work.
b. Find the inverse function, C -1 (x).
c. How long did the technician work on
the furnace if the total charge was
$640?
4. FLOORING Kara is having baseboard
installed in her basement. The total
cost C(x) in dollars is given by
C(x) = 125 + 16x, where x is the
number of pieces of wood required
for the installation.
a. Find the inverse function C -1 (x).
b. If the total cost was $269 and each
piece of wood was 12 feet long, how
many total feet of wood were
used?
5. BOWLING Libby’s family went bowling
during a holiday special. The special cost
$40 for pizza, bowling shoes, and
unlimited drinks. Each game cost $2.
How many games did Libby bowl if the
total cost was $112 and the six family
members bowled an equal number of
games?
4-7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
48
Glencoe Algebra 1
In a function, there is exactly one output for every input. In other words, every element in
the domain pairs with exactly one element in the range. When a function is one-to-one,
each element of the domain pairs with exactly one unique element in the range. When a
function is onto, each element of the range corresponds to an element in the domain.
If a function is both one-to-one and onto, then the inverse is also a function.
Determine whether each relation is a function. If it is a function, determine
if it is one-to-one, onto, both, or neither.
1.
11
16
-3
4
3
6
9
12
2.
1
2
3
4
5
-3
-2
0
4
5
3.
3
6
9
12
15
10
5
0
-5
4.
4
2
7
11
6
1
-2
-4
7
5.
2
6
13
2
3
4
6
8
6.
3
1
-9
10
2
4
11
17
19
Determine whether the inverse of each function is also a function.
7.
y
x
O
8
4
−4
−8
4
8
−4
−8
8.
y
x
O
8
4
−4
−8
4
8
−4
−8
9.
y
x
O
8
4
−4
−8
4
8
−4
−8
Enrichment
One-to-One and Onto Functions
2
6
9
12
-1
3
5
8
9
one–to–one
-3
-2
-1
2
6
3
5
10
onto
5
7
9
10
-6
-11
-15
-19
both
4-7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Assessment
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
Chapter 4
49
Glencoe Algebra 1
Read each question. Then fill in the correct answer.
1. A
B
C
D
2. F
G
H
J
3. A
B
C
D
4. F
G
H
J
5. A
B
C
D
6. F
G
H
J
Multiple Choice
Student Recording Sheet
Use this recording sheet with pages 280–281 of the Student Edition.
Short Response/Gridded Response
Record your answer in the blank.
For gridded response questions, also enter your answer in the grid by writing
each number or symbol in a box. Then fill in the corresponding circle for that
number or symbol.
7.
8.
(grid in)
9.
10a.
10b.
11a.
11b.
11c.
8.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
.
.
.
.
.
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
Extended Response
Record your answers for Question 12 on the back of this paper.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
50
Glencoe Algebra 1
4
Rubric for Scoring Extended Response Test
General Scoring Guidelines
•
If a student gives only a correct numerical answer to a problem but does not show how
he or she arrived at the answer, the student will be awarded only 1 credit. All extended
response questions require the student to show work.
•
A fully correct answer for a multiple-part question requires correct responses for all
parts of the question. For example, if a question has three parts, the correct response to
one or two parts of the question that required work to be shown is not considered a fully
correct response.
•
Students who use trial and error to solve a problem must show their method. Merely
showing that the answer checks or is correct is not considered a complete response for
full credit.
Exercise 12 Rubric
Score
Specifi c Criteria
4
Student explain that the slopes of the lines must be compared. If two lines have
the same slope, they are parallel. If their slopes are opposite reciprocals, they are
perpendicular.
3
A generally correct solution, but may contain minor flaws in reasoning
or computation.
2
A partially correct interpretation and/or solution to the problem.
1
A correct solution with no evidence or explanation.
0
An incorrect solution indicating no mathematical understanding of the concept or
task, or no solution is given.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Assessment
NAME
DATE
PERIOD
Chapter 4
51
Glencoe Algebra 1
4
4
SCORE
For Questions 1 and 2, write an equation in slope-intercept
form for each situation.
1. slope: 1 −
4
, y-intercept: -5
2. line passing through (9, 2) and (-2, 6)
3. Graph 4x + 3y = 12.
4. Write a linear equation in slope-intercept form to model a
tree 4 feet tall that grows 3 inches per year.
5. MULTIPLE CHOICE The table of ordered pairs
shows the coordinates of the two points on
the graph of a function. Which equation
describes the function?
A y = -2x + 1
C y = - 1 −
2
x + 1
B y = 1 −
2
x - 1
D y = - 1 −
2
x - 1
Chapter 4 Quiz 2
(Lessons 4-3 and 4-4)
Chapter 4 Quiz 1
(Lessons 4-1 and 4-2)
x
y
-2
2
4 -1
1. Write an equation in point-slope form for a line that
passes through (3, 6) with a slope of - 1 −
3
.
2. Write y - 9 = -(x + 2) in slope-intercept form.
3. Write an equation in point-slope form for a horizontal
line that passes through (-4, -1).
4. Write an equation in slope-intercept form for the line
that passes through (5, 3) and is parallel to x + 3y = 6.
5. MULTIPLE CHOICE Line DE contains the points D (-1, -4) and
E (3, 3). Line FG contains the point F (-3, 3). Which set of
coordinates for point G makes the two lines perpendicular?
A (1, 7)
C (1, 4)
B (1, 10)
D (4, -1)
SCORE
1.
2.
3.
y
x
O
4.
5.
1.
2.
3.
4.
5.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
NAME
DATE
PERIOD
Chapter 4
52
Glencoe Algebra 1
4
4
SCORE
Chapter 4 Quiz 3
(Lessons 4-5 and 4-6)
1. Find the inverse of {(1, 3), (4, -1), (7, -5), (10, -9)}.
2. Graph the inverse of the function
graphed at the right.
Find the inverse of each function.
3. f (x) = 4x + 6
4. f (x) = 3 −
4
x - 8
5. MULTIPLE CHOICE Write the inverse of
3x + 4y = 12 in f -1 (x) notation.
A f -1 (x) = 12 - 4x −
3
B f -1 (x) = 12 - 3x −
4
C f -1 (x) = 12 - 3x
D f -1 (x) =
12 - 4y
−
3
Chapter 4 Quiz 4
(Lesson 4-7)
For Questions 1–5, use the table.
Age (years)
26
27
28
29
30
Median Income
($1000)
16.8 19.1 23.3 25.8 33.9
1. Make a scatter plot relating age to median income.
Then draw a fit line for the scatter plot.
2. Determine whether the graph shows a positive correlation,
a negative correlation, or no correlation. If there is a positive
or negative correlation, describe its meaning.
3. Write an equation of the best-fit line for the data in the table.
4. Use the line of fit to predict the median income for
32-year olds.
5. MULTIPLE CHOICE What is the correlation coefficient for
the best-fit line?
A 4.09
B –90.74
C 0.943
D 0.971
SCORE
1.
2.
3.
4.
5.
Age (years)
Median Income ($1000)
16
19
22
25
28
31
26
0
27 28 29 30
y
x
O
8
4
−4
−8
4
8
−4
−8
1.
2.
3.
4.
5.
y
x
O
8
4
−4
−8
4
8
−4
−8
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Assessment
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
53
Glencoe Algebra 1
4
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. Which is the slope-intercept form of an equation for the line containing
(0, -3) with slope -1?
A y = -x - 3
B y = -3x - 1
C y = x + 3
D x = -3y - 1
2. Write an equation in slope-intercept form of the line with a slope of - 3 −
4
and y-intercept of – 5.
F y = 5x - 3 −
4
G 3x + 4y = 20
H y = - 3 −
4
x - 5
J y = - 3 −
4
x + 5
3. Write an equation of the line that passes through (-2, 8) and (-4, -4).
A y = 2x + 12
B y = 6x + 20
C y = -6x - 4
D y = 1 −
6
x + 25 −
3
4. Write y - 3 = 2 −
3
(x - 2) in standard form.
F 2x - 3y = 5
G y = 2 −
3
x + 5 −
3
H -2x + 3y = -5 J 2x - 3y = -5
5. Write y - 1 = 2 (x -
3 −
2
) in slope-intercept form.
A 2x - y = 2
B 1 −
2
y + 1 −
2
= x
C y = 2x - 1 −
2
D y = 2x - 2
6. A cell phone company charges $42 per month of service. The cost of a new
cell phone, plus 8 months of service, is $415.99. How much does it cost to
buy a new cell phone and 3 months of service?
F $79.99
G $126.00
H $205.99
J $289.99
Part II
For Questions 7–10, use the following information.
Nikko needs to get his air-conditioner fixed. The technician will charge Nikko a flat fee of
$50 plus an additional $20 for each hour of work.
7. Write an equation to represent Nikko’s total cost to repair
his air-conditioner. Use t for total cost and h for hours.
8. Graph this equation.
9. How much will it cost Nikko if the technician
has to spend 4 hours working on the air-conditioner?
10. How many hours must the technician work for it to
cost Nikko $180?
1.
2.
3.
4.
5.
6.
Assessment
Chapter 4 Mid-Chapter Test
(Lessons 4-1 through 4-3)
Part I
7.
8.
9.
10.
1
Hours
2 3 4 5 6 7 8 9 10
20
40
60
80
100
Total Cost ($)
120
140
160
180
200
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
54
Glencoe Algebra 1
4
SCORE
Choose from the terms above to complete each sentence.
1. If two lines have slopes that are negative reciprocals of each
other, then they are
.
2. A(n)
is the set of ordered pairs obtained
by exchanging the x-coordinates with the y-coordinates of
each ordered pair of a relation or function.
3. A graph of data points is sometimes called a
.
4. If two lines have slopes that are the same, then they are
.
5. The number that describes how closely a best-fit line models a
set of data is called the
.
6. The leftmost data point in a set is (3, 27) and the rightmost
point is (12, 13). If you use a linear prediction equation to find
the corresponding y-value for x = 10, you are using a method
called
.
7. The leftmost data point in a set is (1997, 24) and the rightmost
point is (2011, 38). If you use a linear prediction equation to
find the corresponding y-value for x = 2012, you are using a
method called
.
8. The equation y = -3x + 12 is written in
form.
9. The equation y + 6 = 2(x - 4) is written in
form.
Define each term in your own words.
10. line of fit
11. linear extrapolation
Chapter 4 Vocabulary Test
best-fi t line
bivariate data
correlation coefficient
inverse function
inverse relation
linear extrapolation
linear interpolation
linear regression
median-fi t line
line of fi t
parallel lines
perpendicular lines
point-slope form
rate of change
scatter plot
slope-intercept form
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Assessment
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
55
Glencoe Algebra 1
4
SCORE
Chapter 4 Test, Form 1
Write the letter for the correct answer in the blank at the right of each question.
For Questions 1–5, find the equation in slope-intercept form that
describes each line.
1. a line with slope -2 and y-intercept 4
A y = -2x
B y = 4x - 2
C y = -2x + 4
D y = 2x - 4
2. a line through (2, 4) with slope 0
F y = 2
G x = 2
H y = 4
J x = 4
3. a line through (4, 2) with slope 1 −
2
A y = - 1 −
2
x
B y = 1 −
2
x - 4
C y = 2x - 10
D y = 1 −
2
x
4. a line through (-1, 1) and (2, 3)
F y = 2 −
3
x + 5 −
3
G y = - 2 −
3
x + 5 −
3
H y = 2 −
3
x - 5 −
3
J y = - 2 −
3
x - 5 −
3
5. the line graphed at the right
A y = 2 −
3
x - 1
C y = 2 −
3
x + 3 −
2
B y = 3 −
2
x - 1
D y = 3 −
2
x + 3 −
2
6. If 5 deli sandwiches cost $29.75, how much will 8 sandwiches cost?
F $37.75
G $29.75
H $47.60
J $0.16
7. What is the standard form of y - 8 = 2(x + 3)?
A 2x + y = 14
B y = 2x + 14
C 2x - y = -14
D y - 2x = 11
8. Which is the graph of 3x - 4y = 6 ?
F
y
x
O
G
y
x
O
H
y
x
O
J
y
x
O
9. Which is the point-slope form of an equation for the line that passes through
(0, -5) with slope 2?
A y = 2x - 5
B y + 5 = 2x
C y - 5 = x - 2 D y = 2(x + 5)
10. What is the slope-intercept form of y + 6 = 2(x + 2)?
F y = 2x - 6
G y = 2x - 2
H y = 2x + 6
J 2x - y = 6
11. When are two lines parallel?
A when the slopes are opposite
B when the slopes are equal
C when the slopes are positive
D when the product of the slopes is -1
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
y
x
O
(3, 1)
(0, -1)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
56
Glencoe Algebra 1
4
Chapter 4 Test, Form 1 (continued)
12. Find the slope-intercept form of an equation for the line that passes
through (-1, 2) and is parallel to y = 2x - 3.
F y = 2x + 4
G y = 0.5x + 4
H y = 2x + 3
J y = -0.5x - 4
13. Find the slope-intercept form of an equation of the line perpendicular to
the graph of x - 3y = 5 and passing through (0, 6).
A y = 1 −
3
x - 2
B y = -3x + 6
C y = 1 −
3
x + 2
D y = 3x - 6
For Questions 14 and 15, use the scatter plot shown.
14. How would you describe the relationship between
the x- and y-values in the scatter plot?
F strong negative correlation
G weak negative correlation
H weak positive correlation
J strong positive correlation
15. Based on the data in the scatter plot, what
would you expect the y-value to be for x = 2020?
A greater than 80
C between 65 and 50
B between 80 and 65
D less than 50
16. Which equation has a slope of 2 and a y-intercept of -5?
F y = -5x + 2
G y = 5x + 2
H y = 2x + 5
J y = 2x - 5
17. Which correlation coefficient corresponds to the best-fit line that most closely
models its set of data?
A 0.84
B 0.13
C -0.87
D -0.15
18. The table below shows Mia’s bowling score each week she participated in a
bowling league.
Week
1
2
3
4
5
6
Score
122 131 130 133 145 139
Use the median-fit line to estimate Mia’s score for week 16.
F 173
G 180
H 182
J 257
19. If f(x) = 6x + 3, find f -1 (x).
A f -1 (x) = 6x - 3 B f -1 (x) = x - 6 −
3
C f -1 (x) = x - 3 −
6
D f -1 (x) = -3 - 6x
20. If f(x) = 4(3x - 5), find f -1 (x).
F f -1 (x) = x + 5 −
12
G f -1 (x) = x + 20 −
12
H f -1 (x) = x - 20 −
12
J f -1 (x) = x + 5 −
4
Bonus Find the value of r in (4, r), (r, 2) so that the slope of the
line containing them is - 5 −
3
.
0
50
60
70
80
90
'90 '95 '00 '05 '10
B:
12.
13.
14.
15.
16.
17.
18.
19.
20.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Assessment
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 4
57
Glencoe Algebra 1
4
SCORE
Assessment
Write the letter for the correct answer in the blank at the right of each question.
1. What is the slope-intercept form of the equation of a line with a slope of 5
and a y-intercept of -8?
A y = -8x + 5
B y = 8x - 5
C 5x - y = - 8
D y = 5x - 8
2. Which equation is graphed at the right?
F 2y - x =10
H 2x - y = 5
G 2x + y = -5
J 2y + x = -5
3. Which is an equation of the line that passes
through (2, -5) and (6, 3)?
A y = 1 −
2
x - 6
C y = 2x + 12
B y = 1 −
2
x
D y = 2x - 9
4. What is an equation of the line through (0, -3) with slope 2 −
5
?
F -5x + 2y = 15
H 2x - 5y = 15
G -5x - 2y = -15
J -2x + 5y = 15
5. Which is an equation of the line with slope -3 and a y-intercept of 5?
A y = -3(x + 5)
B y - 5 = -3x
C -3x + y = 5
D y = 5x - 3
6. What is the equation of the line through (-2, -3) with a slope of 0?
F x = -2
G y = -3
H -2x - 3y = 0
J -3x + 2y = 0
7. Find the slope-intercept form of the equation of the line that passes through
(-5, 3) and is parallel to 12x - 3y = 10.
A y = -4x - 17
B y = 4x - 13
C y = - 4x + 13 D y = 4x + 23
8. If line q has a slope of - 3 −
8
, what is the slope of any line perpendicular to q?
F - 3 −
8
G 3 −
8
H 8 −
3
J - 8 −
3
9. A line of fit might be defined as
A a line that connects all the data points.
B a line that might best estimate the data and be used for predicting values.
C a vertical line halfway through the data.
D a line that has a slope greater than 1.
10. A scatter plot of data comparing the number of years since Holbrook High
School introduced a math club and the number of students participating
contains the ordered pairs (3, 19) and (8, 42). Which is the slope-intercept
form of an equation for the line of fit?
F y = 4.6x + 5.2
G y = 3x + 1
H y = 5.2x + 4.6
J y = 0.22x - 1.13
11. Use the equation from Question 10 to estimate the number of students who
will be in the math club during the 15th year.
A 53
B 61
C 65
D 74
Chapter 4 Test, Form 2A
y
x
O
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.