Grade 10 TAKS Mathematics

Objective 1: The student will describe functional relationships in a variety of ways.

For this objective you should be able to recognize that a function represents a dependence of one quantity on another and can be described in a number of ways.

What Is a Function?

A function is a set of ordered pairs ( x , y ) in which each x -coordinate is paired with only one y -coordinate. In a list of ordered pairs belonging to a function, no x -coordinate is repeated.

When people eat at a restaurant, they often use a tip chart to determine the amount of the tip to leave the server. A tip chart
is a good example of a function.
Tip Chart

On the tip chart above, each meal cost listed has exactly one recommended tip listed. Since the amount of the tip depends on
how much the meal costs, the recommended tip is a function of the cost of the meal.

Cost Recommended of Meal Tip
$ 5.00 $ 1.00
$ 6.00 $ 1.20
$ 7.00 $ 1.40
$ 8.00 $ 1.60
$ 9.00 $ 1.80
$ 10.00 $ 2.00

Do you see that . . .
In a functional relationship, for any given input there is a unique output.
If you are given an x -value belonging to a function, you can nd the corresponding y -value.
If you input $ 5.00 into the above function, the output will be $ 1.00.

x
y Function

Input value
Input an x -value and you get a y -value.
Output value 13
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Objective 1
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MAT I CS
There are two ways to test a set of ordered pairs to see whether it is a function.
Examine the list of ordered pairs.
If a set of ordered pairs is a function, no x -coordinate in the set is repeated. No x -coordinate should be listed with two different

y -coordinates.

Is this set of ordered pairs a function?
{ ( 1, 4) , ( 5, 7) , ( 1, 7) , ( 10, 12) }
Examine the set of ordered pairs.
None of the x -coordinates in the set are repeated.

Two ordered pairs, ( 5, 7) and ( 1, 7) , have the same
y -coordinate but different x -coordinates. This does not prevent this set of ordered pairs from being a functional relationship.

This set of ordered pairs is a function.

Examine a graph of the function.
Use a vertical line to determine whether two points have the same x -coordinate. If two points in the function lie on the same vertical line,

then they have the same x -coordinate, and the set of ordered pairs is not a function.

Is this set of ordered pairs a function?
{ ( 2, 5) , ( 0, 7) , ( 1, 4) , ( 2, 6) }
The number 2 is paired with 5; 0 is paired with 7; 1 is paired
with 4; and 2 is paired with 6.

Two ordered pairs, ( 2, 5) and ( 2, 6) , have the same
x -coordinate. In a functional relationship, no x -coordinate should repeat.

This set of ordered pairs is not a function.

Do you see that . . . 14
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In a function, the y -coordinate is described in terms of the x -coordinate. The value of the y -coordinate depends on the value of the x -coordinate.
Do you see that . . .
Suppose the number of miles you walk is equal to 4 times the number of hours you walk. Which is the dependent quantity in
this function?
If you walked for 1 hour, you would have walked 4 miles. If you walked for 3 hours, you would have walked 12 miles.

The distance you walk depends on, or is described in terms of, the number of hours you walk.
In this function, the number of hours you walk is the independent quantity. The distance you walk is the dependent quantity.
An equation that describes this function is d 4 h , where d represents the number of miles you walk, and h represents the
number of hours. In this equation, d , the distance you walk, depends on h , the number of hours you walk.

The variable h is the independent variable.
The variable d is the dependent variable.
The number 4 is a constant , a quantity in an equation that does
not change.

Do the ordered pairs graphed below represent a function?
The ordered pairs ( 2, 5) and ( 2, 7) lie on a common vertical line.
They have the same x -coordinate, 2, but different y -coordinates, 5 and 7.

This graph does not represent a function because two points lie on the same vertical line.

y
x
5
4

6
7
8
9

3
2
1
0
1

2
3
4
5
6
7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 89 15
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MAT H E
MAT I CS Suppose the equation c 0.07 m 0.25 describes c , the cost of a phone call, in terms of m , the number of minutes the phone call lasts. In this function, c is the dependent variable, m is the independent variable, and 0.07 and 0.25 are constants.

Jeremy works at an appliance store. He is paid $ 180.00 a week for his base salary plus a commission equal to 5% of his total sales.
The equation s 180 0.05 d represents Jeremy s weekly salary, s , in terms of d , his total weekly sales in dollars.

Which variable is the dependent variable in this equation? What are the constants?
Jeremy s weekly salary, s , is the dependent quantity because it
depends on d , his total sales.

The constants are 180, Jeremy s base salary, and 0.05,
his commission, because these numbers do not change.

Try It Cara buys milk for her scout camp each morning. The function
below shows the relationship between c , the total cost of the milk she buys, and n , the number of quarts she purchases.

c 1.25 n
In this functional relationship, which value is the dependent quantity?
The _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ quantity is the number of quarts of milk Cara purchases.

The _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ quantity is the total cost of the milk because the cost depends on the number of quarts of
milk Cara buys.

The independent quantity is the number of quarts of milk Cara purchases.
The dependent quantity is the total cost of the milk because the cost
depends on the number of quarts of milk Cara buys. 16
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How Can You Represent a Function?
Functional relationships can be represented in a variety of ways.

Method Description Example
List

Table

Mapping
Description
Equation
y x 1

f ( x ) x 1
Graph y

x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 1 2 345

Graph the ordered pairs.

Write a special type of
equation that uses f ( x )
to represent y .

Function
notation

Write an equation
that describes the
y -coordinate in terms
of the x -coordinate.

The y -values for a set of points are 1 more
than the corresponding x -values.
Use words to describe
the functional
relationship.

3
1
4
10.5

2
2
5
11. 5

Draw a picture that
shows how the ordered
pairs are formed.

Place the ordered pairs
in a table.

{ (

, ( 1, 2) , ( 4, 5) ,
( 10.5, 11.5) , }
List the ordered pairs.

xy



12
45
10.5 11.5
17
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Objective 1
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To use function notation to describe a function, give the function a name, typically a letter such as f , g , or h . Then use an algebraic
expression to describe the y -coordinate of an ordered pair.

MAT H E

MAT I CS Suppose f ( x ) 3 x 1.
This function is read as f of x equals 3 times x minus 1.
If you input x , the output will be 3 x 1.

The y -coordinate of the ordered pair is 3 x 1.

The function described by f ( x ) 3 x 1 is the same as the function described by y 3 x 1. In this function, an ordered

pair looks like this: ( x , 3 x 1) .

Here are three methods you can use to determine whether two different representations of a function are equivalent.

Method
Match a list or table of ordered
pairs to a graph.

Match an equation to a graph.

Match a verbal description to
a graph, an equation, or an
expression written in function
notation.

Action
Show that each ordered pair
listed matches a point on the
graph.

Determine whether they
are both linear or quadratic
functions.

Find points on the graph and
show that their coordinates
satisfy the equation.

Find points that satisfy the
equation and show that they
are on the graph.

Use the verbal description to
nd ordered pairs belonging
to the function and then show
that they satisfy the graph,
equation, or function rule.

Find points on the graph or
ordered pairs satisfying the
equation or rule and show
that they satisfy the verbal
description.

Do you see that . . . 18
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Any equation of the form y mx b is a linear function . Its graph will be a line.
Any equation of the form y ax 2 bx c is a quadratic function . Its graph will be a parabola.
y

x
5
4

6
7
8

9
10
11
12

3
2
1
0

1
2

3
4
5
6

7
8
9
10
11

12

1 1 2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 11 12

xy
0
1
3
5
6

4
1

1
5

4

f ( x ) = x 2 6 x + 4

y
x
5
4

6
7
8

9
10
11
12

3
2
1
0

1
2

3
4
5
6

7
8
9
10
11

12

1 1 2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 11 12
f ( x ) = 3 x 5
xy

0
3
5

5
2 11

10
4 19
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Objective 1
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MAT H E
MAT I CS Which ordered pair, ( 7, 11) or ( 4, 1) , belongs to the function in which the y -coordinate is 3 more than the x -coordinate? Match the ordered pairs to the verbal description. For the ordered pair ( 7, 11) , the y -coordinate should be 3 more than 7.
7 3 10, not 11
The ordered pair ( 7, 11) does not belong to this function.
For the ordered pair ( 4, 1) , the y -coordinate should be
3 more than 4.

4 3 1
The ordered pair ( 4, 1) belongs to this function.

A variety of methods of representing a function are shown below. Which of these examples represents a function that is different
from the other functions?

Look at the ordered pairs that make up each function.
In Examples A, B, C, E, and F, each x -coordinate is paired with
a y -coordinate that is its double, or two times as great.

In Example D , for each ordered pair listed, the x -coordinate is
paired with a y -coordinate that is two more rather than two times as great. So Example D includes ordered pairs that are

not in the other examples.
Only Example D represents a function that is different from the other functions shown.

A. Verbal Description The value of y is double the
value of x .
B. List of Selected Values { ( 3, 6) , ( 2, 4) , ( 1, 2) ,

( 0, 0) , ( 5, 10) , }
C. Table of Selected Values

D. Mapping of Selected Values
E. Graph
F. Equation y 2 x
xy
1 2
24
00
510

5
3
1
0

7
1
3
2 . . . . . .

6
6
6

y
x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 6 1 2 345 20
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The table below presents selected values in a functional relationship.
Write an equation that describes this functional relationship.
Look for a pattern in the ordered pairs that belong to
the function.
The y -coordinate appears to be 5 more than the x -coordinate, so x 5 y should represent the pattern.

Check this equation for each pair.
x 5 y
For x 2 2 5 3
For x 11 5 6
For x 55 5 10
For x 77 5 12

The y -coordinate in each ordered pair is 5 more than the x -coordinate.

The rule for this function can be represented by the equation x 5 y or by the equation y x 5.

x 2157
y 361012

The table on the right presents selected values in a functional relationship between x and y .
Using function notation, write a rule that represents the relationship.
Look for a pattern in the function s ordered
pairs.
The y -coordinate appears to be 5 times the x -coordinate plus 2.

Check this pattern for each pair.
5 x 2 y
( 1, 3) 5 1 2 3
( 1, 7) 5 1 2 7
( 3, 17) 5 3 2 17
( 8, 42) 5 8 2 42

The y -coordinate is equal to 5 times the x -coordinate plus 2.
The rule for this function can be represented by the equation y 5 x 2.

Replace y with f ( x ) to express the rule in function notation: f ( x ) 5 x 2. Do you see
that . . .

For any equation A B , it is also true that
B A . The function
y 2 x 1 is the
same as the function 2 x 1 y .

xy
1 3
17
317
842 21
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MAT H E
MAT I CS Does the graph below represent the same function as the equation y 3 x 2 1?

The function y 3 x 2 1 is a quadratic function because there
is an x 2 term in its equation. Its graph should be a parabola. The above graph is a parabola, so the equation and the graph

are the same type of function.
Show that the equation and the graph represent the same
quadratic function with the same set of ordered pairs.

First check the coordinates of points on the graph to see
whether they satisfy the equation. Pick points on the graph whose coordinates are easy to read. For example, you might

choose ( 0, 1) , ( 1, 2) , and ( 2, 11) .
Substitute these values into y 3 x 2 1 and determine whether the equation is true.

Point ( 0, 1) Point ( 1, 2) Point ( 2, 11) x 0 and y 1 x 1 and y 2 x 2 and y 11
y 3 x 2 1 y 3 x 2 1 y 3 x 2 1 1
3( 0) 2 12
3( 1) 2 111
3( 2) 2 1
1
3( 0) 12
3( 1) 111
3( 4) 1 1
0 12
3 111
12 1
1 12 211 11
The points ( 0, 1) , ( 1, 2) , and ( 2, 11) are points on the graph, and their coordinates satisfy the equation.

Next nd ordered pairs that satisfy the equation and con rm
that the points are on the graph. Pick values that are easy to substitute, like x 1, x 0, or x 2, and nd the

corresponding values for y .

y
x
3
2
1
0

1
2

3
4
5
6

7
8
9
10

13
14

11
12

1 1 2 3 4 5 6 2 3 4 5 6
( 2, 11)
( 1, 2)
( 0, 1) 22
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Try It The equation y x 2 1 represents a functional relationship.
Which graph represents this function?

Determine whether the equation is a linear or quadratic function.
The equation is a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ function because it contains the term x 2 , which has an exponent of 2.

Its graph must be a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .

y
x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 1 2 345

y
x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 1 2 345

y
x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 1 2 345

y
x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 1 2 345

Substitute these values into y 3 x 2 1 and determine the value for y .
x 1 x 0 x 2
y 3 x 2 1 y 3 x 2 1 y 3 x 2 1 y 3( 1) 2 1 y 3( 0) 2 1 y 3( 2) 2 1

y 3( 1) 1 y 3( 0) 1 y 3( 4) 1 y 3 1 y 0 1 y 12 1
y 2 y 1 y 11
The ordered pairs ( 1, 2) , ( 0, 1) , and ( 2, 11) satisfy the equation.

Con rm that these ordered pairs are points on the graph. Yes, all three points are on the graph.
The graph does represent the relationship y 3 x 2 1.

A C
B D 23
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Objective 1
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MAT H E
MAT I CS Answer choices _ _ _ _ _ _ _ and _ _ _ _ _ _ _ cannot be the graph of this function because they are _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . Determine which parabola is the correct graph. See whether the point ( 0, 1) in answer choice B satis es the
equation y x 2 1.
When x 0 and y _ _ _ _ _ _ _ , is the equation y x 2 1 true?
Does _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ?
No, _ _ _ _ _ _ _ _ _ _ _ _ _ _ .
Answer choice B is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .

See whether the point ( 0, 1) in answer choice C satis es the
equation y x 2 1.
When x _ _ _ _ _ _ _ and y _ _ _ _ _ _ _ , is the equation y x 2 1
true?
Does _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ?
Yes, _ _ _ _ _ _ _ _ _ _ _ _ _ _ .
Answer choice C is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .

The equation is a quadratic function because it contains the term x 2 , which
has an exponent of 2. Its graph must be a parabola. Answer choices A and
D cannot be the graph of this function because they are lines. When x 0
and y 1, is the equation y x 2 1 true? Does 1 0 2 1? No, 1 1.
Answer choice B is not correct. When x 0 and y 1, is the equation y
x 2 1 true? Does 1 0 2 1? Yes, 1 1. Answer choice C is correct. 24
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How Can You Draw Conclusions from a Functional Relationship?
Use these guidelines when interpreting functional relationships in a real-life problem.

Understand the problem.
Identify the quantities involved and any relationships between them.
Determine what the variables in the problem represent.
For graphs: Determine what quantity each axis on the graph
represents. Look at the scale that is used on each axis.

For tables: Determine what quantity each column in the table
represents.

Look for trends in the data. Look for maximum and minimum
values in graphs.

Look for any unusual data. For example, does a graph start at
a nonzero value? Is one of the problem s variables negative at any point?

Match the data to the equations or formulas in the problem.

The graph below shows the daily high temperatures in degrees Fahrenheit over a two-week period in August. During which
three-day period did the temperature decrease by the greatest number of degrees?

The temperature decreased from August 14 to August 16 and then again from August 17 to August 21. To determine which three-day
period it decreased by the greatest number of degrees, you need to nd the coordinates of these points and calculate the total drop in
the daily high temperature.
You could build a table to organize your work.

Temperature
( ° F)

Daily High Temperatures

Date in August
791113 15 17 19 810121416182021

60
65
70
75
80
85
90
95
100
105

0 25
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Objective 1 MAT
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The high temperature decreased by 18 ° F from August 19 to August 21. This was the greatest decrease in temperature for any three-day
period on the graph.

3-Day Day 1 High Day 2 High Day 3 High Decrease
Period Temperature Temperature Temperature

Aug. 14, 15, 16 105 ° F 103 ° F 100 ° F 5 ° F
Aug. 17, 18, 19 102 ° F 100 ° F 98 ° F 4 ° F
Aug. 18, 19, 20 100 ° F 98 ° F 85 ° F 15 ° F
Aug. 19, 20, 21 98 ° F 85 ° F 80 ° F 18 ° F

25
The formula for converting degrees Celsius, C, to degrees
Fahrenheit, F, is F 9 5 C 32.

Does the graph represent this function accurately?
The points increase at a rate of 9 ° F for every 5 ° C, and the slope

of the graph should be 9 5 , the coef cient of C.
Two points on the graph are ( 0, 32) and ( 100, 212) . When the

coordinates are substituted into the equation F 9 5 C 32, the
equation is true.

F 9 5 C 32 F 9 5 C 32

212 9 5 ( 100) 32 32 9 5 ( 0) 32
212 180 32 32 0 32 212 212 32 32

Yes, the graph accurately represents this function.

90 100 110 0
20
40
60
80
100
120
140
160
180
200
220

10 20 30 40 50 60 70 80
Degrees Celsius

Degrees
Fahrenheit

Now practice what you ve learned. 26
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Question 1
Rhonda works at a grocery store after school. She is paid $ 5.50 per hour. Her weekly salary, s ,

is described by the function s 5.5 h , where h is the number of hours she works in a week. What
is the dependent quantity in this functional relationship?

A The number of hours she works in a week
B The number of dollars she is paid per hour
C The total salary for a week
D The number of days she works in a week

Question 2
The number of pretzels, p , that can be packaged in a box with a volume of V cubic units is given

by the equation p 45 V 10. In this relationship, which is the dependent variable?

A 10
B 45
C p
D V

Question 3
Which of the following tables does not represent a function?

Question 4
The table shows the independent and dependent values in a functional relationship. Which

function best represents this relationship?

A f ( x ) 2 x 2 2 x 1
B f ( x ) 2 x 2 2 x 1
C f ( x ) x 2 1
D f ( x ) 8 x 9

Answer Key: page 224
Answer Key: page 224

Answer Key: page 224 Answer Key: page 224
Independent Dependent
01
15
213
325

AC
BD

xy
12
41
12
41

xy
1 1
33
22
55

xy
11
24
39
116

xy
20
11
35
40 27
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MAT H E
MAT I CS Question 5 Jane started a weekend pet-care business. She bought the necessary supplies for $ 210. Jane charges $ 25 per weekend for each pet she cares for. Which function best represents her net pro t
in terms of x , the number of pets she cares for?
A f ( x ) x 25 210
B f ( x ) 210 25 x
C f ( x ) 210 x 25
D f ( x ) 25 x 210

Question 6
Jeb s stereo is playing at a volume of 75 decibels. If the decibel level reaches 120 decibels, the

neighbors will complain. Which inequality models q , the number of decibels Jeb can increase
the volume before the neighbors complain?
A 75 q 120
B 75 q 120
C 75 q 120
D 75 q 120

Question 7
Which table best represents the function
f ( x ) 2 5 x 3?

Answer Key: page 224 Answer Key: page 224
Answer Key: page 224
AC

BD

xy
0 3
5 1
10 1
15 9

xy
10 7
0 3
10 1
20 5

xy
1 5
10 1
30 9
20 5

xy
10 7
30 9
60 21
15 3 28
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Question 8
Which graph best represents the function f ( x ) x 2 9?

AC

BD
Answer Key: page 225

y
x
5
4

6
7
8
9

3
2
1
0
1

2
3
4
5
6
7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

y
x
5
4

6
7
8
9

3
2
1
0
1

2
3
4
5
6
7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

y
x
5
4

6
7
8
9

3
2
1
0
1

2
3
4
5
6
7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

y
x
5
4

6
7
8
9

3
2
1
0
1

2
3
4
5
6
7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 29
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MAT H E
MAT I CS Question 9 In chemistry lab Sonya places an empty beaker with a mass of 35 grams on a scale. She adds 5-gram measures of water to the beaker until the total mass of the beaker and water is 85 grams. Each time Sonya adds 5 grams of water, she records the mass. Which graph best illustrates the combined mass of the beaker and water as the amount of water in the beaker increases?

AC

BD
Answer Key: page 225

45 0
10
20
30
40
50
60
70
80

510152025303540
Water Added
( g)

Total
Mass
of Beaker
with Water

0
10
20
30
40
50
60
70
80

Water Added
( g)

Total
Mass
of Beaker
with Water

9 1 2 3 4 5 678

45 0
10
20
30
40
50
60
70
80

510152025303540
Water Added
( g)

Total
Mass
of Beaker
with Water

9 0
10
20
30
40
50
60
70
80

1 2 3 4 5 678
Water Added
( g)

Total
Mass
of Beaker
with Water 30
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Question 10
Jesse sells peanut butter cookies and chocolate chip cookies to raise money for his club. On average, he sells about twice as many chocolate chip cookies as he does peanut butter cookies. Which graph

best models this relationship?

AC

BD
Answer Key: page 225

20 0
4
8
12
16
20
24
28
32

481216
Peanut Butter Cookies

Chocolate
Chip
Cookies

20 0
4
8
12
16
20
24
28
32

481216
Peanut Butter Cookies

Chocolate
Chip
Cookies

20 0
4
8
12
16
20
24
28

481216
Peanut Butter Cookies

Chocolate
Chip
Cookies

20 0
4
8
12
16
20
24
28

481216 24
Peanut Butter Cookies

Chocolate
Chip
Cookies 31
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MAT H E
MAT I CS Question 11 Members of the junior class are raising money for their class trip by selling artwork by local artists. They bought the merchandise from the artists for a total price of $ 1025. The graph
shows their cumulative total income for the two weeks of the sale. On what day of the sale did
they rst make a pro t?

A Day 5
B Day 4
C Day 0
D Day 14

Question 12
Potassium nitrate dissolves in water to form a solution. The graph below shows the solubility of

potassium nitrate in water as a function of the temperature of the water.

According to this graph, about how many grams of potassium nitrate will dissolve in 100 cm 3 of
water at 42 ° C?
A 58 grams
B 82 grams
C 77 grams
D 68 grams

90 10 20 30 40 50 60 70 80
Temperature of Water
( ° C)

Solubility
of
Potassium

Nitrate

( g/
100

cm
3
)

0
20
40
60
80

100
120
140
130

110
90
70
50

30
10
910111213 0
2
4
6
8
10
12
14
16
18
20
22

1 2 3 4 5 678
Days

Total Sales
( hundreds
of dollars)

Answer Key: page 225 Answer Key: page 226 32
32 Page 33 34
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Question 13
A home-builders group recently published a study comparing the cost of building homes from 1,000 to 3,000 square feet in area in four different communities. The study found that the formulas below

predicted the approximate cost, c , of building a new home in each of these communities in terms of f , the area of the home in square feet.

Based on these formulas, in which community would it cost the least to build a home with an area of 1,450 square feet?
A T
B S
C R
D V

Community Cost of Building a New Home
R c 15,000 80 f
S c 25,000 75 f
T c 60,000 50 f
V c 40,000 65 f

Answer Key: page 226 33
33 Page 34 35
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33

The student will demonstrate an understanding of the properties and attributes
of functions.

Objective 2

For this objective you should be able to
use the properties and attributes of functions;

use algebra to express generalizations and use symbols to
represent situations; and

manipulate symbols to solve problems and use algebraic skills to
simplify algebraic expressions and solve equations and inequalities in problem situations.

What Are Parent Functions?
The simplest linear function, y x , is the linear parent function .

If the graph of any function is a line, then its parent function
is y x .

A linear equation never has variables raised to a power other
than 1.

If an equation is linear, then its parent function is y x .

y
x
4
3
2
1

1
2
3
4

0 1 2 3 4 1 2 3 4
y =
x 34
34 Page 35 36
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What is the parent function of this graph?

Since the graph of y x 5 is a line, its parent function is the linear parent function, y x .
x
4
5

3
2
1

1
2
3
4
5

0 1 2 3 4 5 1 2 34
y =
x

y =
x

5

Parent
function

5

x
4
5

3
2
1

1
2
3
4
5

0 1 2 3 4 5 1 2 3 4
y =
x

5

5

y 35
35 Page 36 37
Objective 2
35
MAT H E
MAT I CS

The simplest quadratic function, y x 2 , is the quadratic parent function .
If the graph of any function is a parabola, then its parent function
is y x 2 .

If an equation can be written in the form y ax 2 bx c , then
it is quadratic.

If an equation can be written in this form, then its parent
function is y x 2 .

y
x
4
3
2
1

1
2
3
4

0 1 2 3 4 1 2 3 4
y = x 2

MAT H E
MAT I CS What is the parent function of the equation y
1
2 x 1?
Since the equation y 1 2 x 1 is a linear equation, its graph is a

line. Its parent function is the linear parent function, y x .

y

x
4
3
2
1

1
2
3
4

0 1 2 3 4 1 2 3 4
y =
x

y =
x
1

Parent
function

1 2 36
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What is the parent function of this graph?
Since the graph of y x 2 3 is a parabola, its parent function is the quadratic parent function, y x 2 .
y

x
6
7
8

1
2
3

5
4

0 1 2 3 4 1 2 34

y = x 2 + 3
y = x 2
Parent
function

y
x
6
7
8

1
2
3

5
4

0 1 2 3 4 1 2 34

y = x 2 + 3 37
37 Page 38 39
What Are the Domain and Range of a Function?
A function is a set of ordered pairs of numbers ( x , y ) such that no x -values are repeated. The domain and range of a function are sets

that describe those ordered pairs.

The domain is the set of all the values of the independent
variable, the x -coordinate.

The range is the set of all the values of the dependent variable,
the y -coordinate.

De nition Example { ( 0,1) , ( 2, 6) , ( 3, 5) }
Domain All the x -coordinates in the { 0, 2, 3} function s ordered pairs
Range All the y -coordinates in the { 1, 5, 6} function s ordered pairs

Objective 2

37
MAT H E
MAT I CS What is the parent function of y x
2 ?

The equation y x 2 is a quadratic equation; therefore, its parent function is the quadratic parent function, y x 2 .
y
x
4
3
2
1

1
2
3
4

0 1 2 1 2

Parent
function

y = x 2
y = x 2

Identify the domain and range of the function below.
{ ( 3, 9) , ( 5, 39) , ( 9, 23) , ( 6, 14) }
The domain is the set of x -coordinates in the ordered pairs: { ( 3, 9) , ( 5, 39) , ( 9, 23) , ( 6, 14) } . The domain is { 3, 5, 6, 9} .

The range is the set of y -coordinates in the ordered pairs: { ( 3, 9) , ( 5, 39) , ( 9, 23) , ( 6, 14) } . The range is { 9, 14, 23, 39} .

See Objective 1,
page 12, for more
information about
functions. 38
38 Page 39 40
The domain and range of algebraic functions are usually assumed to be the set of all real numbers. In some cases, however, the domain or
range of a function may be a subset of the real numbers because certain numbers would not make sense in a real-life problem situation.

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Try It What are the domain and range of the function below?
{ ( 4, 9) , ( 5, 16) , ( 6, 25) , ( 7, 36) }
The domain of a function is the set of all _ _ _ _ _ -coordinates.
The domain of this function is { _ _ _ _ _ , _ _ _ _ _ , _ _ _ _ _ , _ _ _ _ _ } .
The range of a function is the set of all _ _ _ _ _ -coordinates.
The range of this function is { _ _ _ _ _ , _ _ _ _ _ , _ _ _ _ _ , _ _ _ _ _ } .

The domain of a function is the set of all x -coordinates. The domain of
this function is { 5, 4, 6, 7} . The range of a function is the set of all
y -coordinates. The range of this function is { 36, 9, 16, 25} .

Consider the function l 4 h , in which l equals the number of legs on h horses. Are there any values that would not be reasonable to
include in the domain or range of this function?
The domain of this function is the set of values you may choose
for h , the independent variable. Would it be reasonable to let h 1.2? No. The variable h represents a number of horses; it

must be a nonnegative integer. The domain is the set of nonnegative integers, { 0, 1, 2, 3, } .

It would not be reasonable to include any other numbers in the domain.
The range of this function is the set of values you will obtain
for the dependent variable, l , the number of legs for a group of h horses. Is it possible to get 6 as a value for l ? Could a group of

horses normally have 6 legs? No, 6 is not a reasonable value for the range of this function. Since 1 horse has 4 legs, 2 horses
have 8 legs, and so on, the range of this function is the set of multiples of 4, or { 0, 4, 8, 12, } .

It would not be reasonable to include any other numbers in the range. 39
39 Page 40 41
Objective 2
39
MAT H E
MAT I CS

The graph of a function also can tell you about its domain and range.
The domain of a function is the set of all the x -coordinates in the
function s graph.

The range of a function is the set of all the y -coordinates in the
function s graph.

91011
19
11
12
13
14
15
16
17
18

9
10

1
2
3
4
5
6
7
8

0 1 2 3 4 5 6 78

y
x
Domain

Range
4 < y < 18

3 < x < 10

The total volume in sales generated by a television commercial is a function of the number of people who watch the commercial.
Wilson Electronics executives estimate that s , the dollars they will generate in sales for the number of people, n , who watch their
commercial, is described by the function s 0.0035 n .
What set of numbers would be an appropriate domain for this function: the integers, the real numbers, the rational numbers, or

the whole numbers?
The domain of this function is the number of people who watch the commercial. The number of people cannot be a fraction or a

negative number. Of the choices given, the only set of numbers that does not contain any fractions or negative numbers is the
whole numbers.
The set of whole numbers would be an appropriate domain for this function. 40
40 Page 41 42
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What inequalities best describe the domain and range of the function represented in this graph?
The domain of this function is the set of x -values in the graph.
The range of this function is the set of y -values in the graph.

Domain
Range
9 10

19
11
12
13
14
15
16
17
18

9
10

1
2
3
4
5
6
7
8

0 1 2 3 4 5 6 78

y
x
0 < y < 16

1 < x < 9

9 10
19
11
12
13
14
15
16
17
18

9
10

1
2
3
4
5
6
7
8

0 1 2 3 4 5 6 78

y
x
An open circle on a graph means that the
point is not included in the solution.

The point ( 5, 3) is not included in the solution.
0
1
2
3
4
5

1 2 345

y
x 41
41 Page 42 43
Objective 2
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MAT H E
MAT I CS In a chemistry experiment two chemicals are poured into a beaker to react. The temperature of the solution is taken every minute for 10 minutes beginning at 1: 05 P. M. The graph below shows the data from this experiment.

What is a reasonable range for this function?
The range of the function is the set of all the y -coordinates in the function. The y -coordinates represent the temperature of the mixed

chemicals. The temperature during the 10-minute interval begins at 10 ° C, and it ends at 50 ° C.

A reasonable range for this function is 10 y 50.

10
20

0

30
40
50
60

1: 06 1: 14 1: 08 1: 10 1: 12
Time

Chemistry Experiment
Temperature ( ° C) P. M. P. M. P. M. P. M. P. M. 42
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Try It The number of pounds of potato salad, p , that will be needed for a
company picnic is given by the function p 0.25 n 4, in which n equals the number of people who will attend the picnic. Each employee in the
company can attend the picnic, and each can bring 3 guests. A total of 12 employees and guests have already signed up to attend the picnic. If
the company employs a total of 40 people, what is a reasonable range for this function?

The range of the function is the set of all the possible values for the
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ variable in the function, the amount of potato salad
to be purchased.

To determine the range of the function, rst determine the minimum and
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ number of people who will attend the picnic.

The minimum number of people is _ _ _ _ _ _ _ _ _ _ , since that many people have
already signed up.
The company has _ _ _ _ _ _ _ _ _ _ employees. If every employee attends and
brings 3 guests, then the maximum number of people is _ _ _ _ _ _ _ _ _ _
because 40 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .
Use the function p _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ to nd
the number of pounds of potato salad that will be needed.
If 12 people attend, then p _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4;
_ _ _ _ _ _ _ _ _ _ pounds of potato salad will be needed.
If 160 people attend, then p _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4;
_ _ _ _ _ _ _ _ _ _ pounds of potato salad will be needed.
The number of pounds of potato salad that will be needed is between
_ _ _ _ _ _ _ _ _ _ pounds and _ _ _ _ _ _ _ _ _ _ pounds.
A reasonable range for this function is _ _ _ _ _ _ _ _ _ _ p _ _ _ _ _ _ _ _ _ _ .

The range of the function is the set of all the possible values for the dependent
variable in the function, the amount of potato salad to be purchased. To
determine the range of the function, first determine the minimum and maximum
number of people who will attend the picnic. The minimum number of people is
12, since that many people have already signed up. The company has
40 employees. If every employee attends and brings 3 guests, then the maximum
number of people is 160 because 40 4 160. Use the function p 0.25 n 4 to
nd the number of pounds of potato salad that will be needed. If 12 people attend,
then p 0.25 12 4; 7 pounds of potato salad will be needed. If 160 people
attend, then p 0.25 160 4; 44 pounds of potato salad will be needed. The
number of pounds of potato salad that will be needed is between 7 pounds and
44 pounds. A reasonable range for this function is 7 p 44. 43
43 Page 44 45
Objective 2
43
MAT H E
MAT I CS
How Can You Interpret a Problem from a Graph?
To interpret a problem situation described in terms of a graph, follow these guidelines.

Identify the quantities that are being compared.
Understand what relationship the graph is describing.
Look at the scales used on the axes of the graph.
Identify the domain or range of the function graphed.
Look for patterns in the data increases, decreases, or data that
remain constant.

Joe walked at a constant speed. The graph represents his distance from his starting point in terms of the time he walked.

Give a reasonable description of the route Joe walked.
The y -axis represents Joe s distance from his starting point. The x -axis represents his walking time. Did Joe get farther and farther

away from his starting point as time went by? No, not for the entire time graphed.

At rst Joe s distance from his starting point was 0. As time went by, this distance increased, but then it returned to 0. He ended his
walk where he began it.
A reasonable description of the route Joe walked is that he started at a certain point, walked for a while in one direction, and then

turned around and returned to his starting point. The graph represents such a route.

Time Joe Walked
Joe s
Distance
from His
Starting
Point

y

x 44
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Try It Te rri placed a pot of water on the stove to boil. The temperature of
the water in terms of the time it was on the stove is represented by the graph.

What is a reasonable interpretation of the graph?
At rst the water temperature _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .
Then the water temperature remained _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ for a while.

Finally the water temperature _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ slowly.
A reasonable interpretation of the graph is that the water
temperature _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ until the water boiled. Then it
remained at a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ temperature until Terri turned
the stove off. Finally it _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ slowly to room
temperature, where it remained constant.

At rst the water temperature increased. Then the water temperature
remained constant for a while. Finally the water temperature decreased
slowly. A reasonable interpretation of the graph is that the water temperature
increased until the water boiled. Then it remained at a constant temperature
until Terri turned the stove off. Finally it cooled slowly to room temperature,
where it remained constant.

Time
Temperature 45
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Objective 2
45
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MAT I CS
What Is a Correlation in a Scatterplot?
One way to represent a set of related data is to graph the data using a scatterplot. In a scatterplot each pair of corresponding values in the

data set is represented by a point on a graph.

To make predictions using a scatterplot, look for a correlation, or pattern , in the data. The pattern may not be true for every point, but look for the
overall pattern the data seem to best t. 46
46 Page 47 48
As you move from left to right on as shown in they show this
the graph, if the this scatterplot type of correlation: data points

positive correlation

negative correlation
no correlation

unde ned correlation
51015 0
5
10
15
20
25

51015 0
5
10
15
20
25

51015 0
5
10
15
20
25

51015 0
5
10
15
20
25

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show a horizontal or vertical pattern
show no pattern
move down
move up 47
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Objective 2
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MAT H E
MAT I CS In a survey of property values, p , the price of an acre of land, was compared to d , the distance of the land from the center of town. The data were graphed in a scatterplot. Describe the correlation between the cost of an acre of land and the land s distance from the center of town.

Look for a pattern in the graph. The general tendency is for the price of an acre of land to decrease as the land s distance from the
center of town increases. The price of an acre of land has a negative correlation to the land s distance from the center of town.

Distance from
Center of Town

Price
of Land
per Acre

Try It Raśl is a sport sherman. He weighs each sh he catches, and he
measures its length. He graphed his data in a scatterplot.

Describe the correlation between the lengths and weights of the sh Raśl caught.
As the lengths of the sh _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ , their weights generally _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .
This is a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ correlation.
As the lengths of the sh increase, their weights generally increase. This is a
positive correlation.

Fish
Weight

Fish Length 48
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How Do You Use Symbols to Represent Unknown Quantities?
Represent unknown quantities with variables , or letters such as x or y . Use variables in expressions, equations, or inequalities.

Jacob, Barbara, and Felix all have part-time jobs after school. Jacob earns $ 5 more per week than Felix, and Barbara earns twice as
much per week as Jacob. The combined earnings of these three students are $ 115 per week. Write an equation that can be used to
nd how much each student earns per week.
Represent the number of dollars each student earns per week.

x Felix s earnings
x 5 Jacob s earnings ( $ 5 more than Felix s earnings)
2( x 5) Barbara s earnings ( twice Jacob s earnings)
Represent the sum of their earnings.

Felix Jacob Barbara
x ( x 5) 2( x 5)
Write an equation setting this sum equal to $ 115.

x ( x 5) 2( x 5) 115

Quattro plans to paint his living room walls. The room is 3 feet longer than it is wide. The walls are 8 feet high. If a can of paint
covers approximately 200 ft 2 , what expression can be used to represent the minimum number of cans of paint Quattro will need?

Represent the area of each wall.

Represent the sum of the areas of the four walls.
2( smaller wall) 2( larger wall)
2( 8 x ) 2 [ 8( x 3) ]
16 x 16( x 3)
16 x 16 x 48
32 x 48
The total area of the four walls is represented by the expression 32 x 48. Since the paint in each can covers 200 ft 2 , divide the area

by 200 to represent the number of cans of paint needed: 32 x 200 48 .

S m a l l e r W a l l s L a r g e r W a l l s
x width of the room x 3 length of the room
8 height of the room 8 height of the room
8 x area 8( x 3) area 49
49 Page 50 51
Objective 2
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MAT H E
MAT I CS
How Do You Represent Patterns in Data Algebraically?
To r epresent patterns in data algebraically, follow these guidelines.
Identify what quantities the data represent.

Identify the relationships between those quantities.
Look for patterns in the data.
Use symbols to translate the patterns into an algebraic
expression or equation.

Ronald and Jaime make weekly deposits into their savings accounts. The table below shows the opening balances and the
balances for each account after the rst four weekly deposits.
Account Balances

If the pattern continues, what expression can be used to represent the balance in Jaime s account after n weeks?
Jaime s opening balance is $ 180. His balance increases by $ 18 each week. His balance in n weeks can be represented by the expression
18 n 180. See whether this expression works for all the known values.

If the pattern continues, the balance in Jaime s account after n weeks can be represented by the expression 18 n 180.

Name Opening Week 1 Week 2 Week 3 Week 4 Balance
Ronald $ 100 $ 124 $ 148 $ 172 $ 196
Jaime $ 180 $ 198 $ 216 $ 234 $ 252

Week 18 n 180 Balance Yes / No ( dollars)
118 1 180 198 198 Yes
218 2 180 216 216 Yes
318 3 180 234 234 Yes
418 4 180 252 252 Yes 50
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The table below represents a functional relationship between x , the lengths of the bases of a series of triangles, and y , their heights. If
the pattern continues, what expression can be used to express their heights in terms of their bases?

The differences between the y -values are not constant. For example, 5 2 3, but 10 5 5. This is not a linear relationship.
To determine whether the relationship is quadratic, compare the y -values to the square of the x -values.

The y -values appear to be 1 more than the squares of the x -values.
See whether all the coordinates satisfy the rule y x 2 1. Substitute the values of x and y from the table into the equation.

All the ordered pairs satisfy the equation y x 2 1.
Since every pair in the table satis es the equation, then y x 2 1 expresses the triangles heights in terms of their bases. .

xy
12
25
310
417

Change in x -values Change in y -values + 1 + 1
+ 1

+ 3
+ 5
+ 7

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xx 2 y
112
245
3910
41617

x y x 2 1 y Yes / No
2
( 1) 2 1 12
1 12Yes
2 2
5
( 2) 2 1 25
4 15Yes

5 5
10
( 3) 2 1 310
9 110 Yes

10 10
17
( 4) 2 1 4 17
16 117 Yes

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Objective 2
51
How Do You Simplify Algebraic Expressions?
You can use the commutative, associative, and distributive properties to simplify algebraic expressions. Use these properties to remove

parentheses and combine like terms.

MAT H E
MAT I CS

Use the commutative property to change the order of the terms
in addition and multiplication.

Use the associative property to change the groupings in addition
or multiplication.

Use the distributive property to remove the parentheses by
multiplying the term outside the parentheses by each term inside the parentheses.

Like terms are terms in an algebraic expression
that use the same variable raised to the
same power.
For example, in the expression 6 t 3 2 t 3 ,

6 t 3 and 2 t 3 are like terms because they are
both expressed in terms of the same variable, t ,
raised to the same power, 3.

Like terms in an algebraic expression
can be combined.
6 t 3 2 t 3 4 t 3

Property
Name ( If a , b , and c are any Examples three real numbers,
then )

a b b a 4 2 x 2 x 4 Commutative
Property or or
ab ba 5 y 2 x 5 xy 2

a ( b c ) ( a b ) c ( 5 y ) 2 y 5 ( y 2 y )
Associative 5 3 y Property or or

a ( bc ) ( ab ) c 5( 3 m ) ( 5 3) m 15 m
Distributive 7( 2 x 3) 7( 2 x ) 7( 3) Property
14 x 21 a ( b c) ab ac 52
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If the length of a rectangle is represented by l and its width is 3 units less than its length, does the expression l 2 3 l represent
its area?
Let l represent the length of the rectangle.

Let l 3 represent the width ( 3 units less than the length) .
Then l ( l 3) represents the area of the rectangle, A lw .

Simplify the expression l ( l 3) .

l ( l 3) l l l 3
l 2 3 l
Yes, the expression l 2 3 l represents the area of the rectangle.

Simplify the expression 5 y 3 3 2 y 3 1.
5 y 3 3 2 y 3 1 ( 5 y 3 2 y 3 ) ( 3 1)
3 y 3 4
When simpli ed, the expression 5 y 3 3 2 y 3 1 3 y 3 4. 53
53 Page 54 55
When simplifying expressions, it is
helpful to remember that the expressions m
and 1 m are equivalent.

Objective 2

53
MAT H E
MAT I CS The lengths of the three sides of a triangle are represented by the expressions 2 m 5, m 1, and 7 m 3. Write an expression in terms of m that can be used to represent the perimeter of the triangle. The perimeter of a triangle is the sum of the lengths of its
three sides. The perimeter of the triangle can be represented by the expression 2 m 5 m 1 7 m 3.
Simplify this expression.
2 m 5 m 1 7 m 3 ( 2 m 1 m 7 m ) ( 5 1 3)
10 m 7
The perimeter of the triangle can be represented by the expression 10 m 7.

Simplify the expression ( 4 b 2 6) ( 2 b 2 3) .
When parentheses are preceded by a negative sign, it means the quantity in parentheses is multiplied by 1.

( 4 b 2 6) ( 2 b 2 3) ( 4 b 2 6) 1( 2 b 2 3)
4 b 2 6 2 b 2 3
( 4 b 2 2 b 2 ) ( 6 3)
2 b 2 3
When simpli ed, the expression ( 4 b 2 6) ( 2 b 2 3) 2 b 2 3.

Do you see that . . . 54
54 Page 55 56
See Objective 5,
page 122, for more
information about
the FOIL method.

Mr. and Mrs. Seymour have a dog pen in their backyard, as shown by the shaded gure in the diagram below.
Write an expression that can be used to represent the area of the yard that does not include the dog pen.
The area of the entire yard, a rectangle, is equal to its width times
its length.

The area of the rectangle is x ( 2 x 6) . Simplify.

x ( 2 x 6) 2 x 2 6 x
The area of the dog pen is equal to its length times its width. The
area of the smaller rectangle is ( x 12) ( x 16) . Simplify by using the FOIL method.

( x 12) ( x 16) ( x x ) ( x 16) ( 12 x ) ( 12 16)
x 2 ( 16 x 12 x ) 192
x 2 28 x 192
Subtract to nd the area of the yard not including the dog pen.

Area of yard Area of pen
( 2 x 2 6 x ) ( x 2 28 x 192)
Simplify by multiplying the quantities in the second parentheses
by 1.

( 2 x 2 6 x ) 1( x 2 28 x 192) 2 x 2 6 x x 2 28 x 192
( 2 x 2 x 2 ) ( 6 x 28 x ) 192
x 2 34 x 192
The expression x 2 34 x 192 represents the area of the yard that does not include the dog pen.

2 x + 6
x 16
x 12 x

Backyard
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Objective 2
55
MAT H E
MAT I CS

Solve the equation 3( y 2) 9.
3( y 2) 9
3 y 3( 2) 9
3 y 6 9
6 6
3 y 15

3
3y
3
15

y 5
In this equation, y 5.

Solve the equation 2 x 5 x 4.
2 x 5 x 4
1 x 1 x
x 5 4
5 5
x 9
In this equation, x 9.

How Do You Solve Algebraic Equations?
To solve algebraic equations, follow these guidelines.
Simplify any expressions in the equation.

Add or subtract on both sides of the equation to get variable
terms on one side and constant terms on the other.

Simplify again if necessary.

Multiply or divide to obtain an equation that has the variable
isolated with a coef cient of 1. In the term 6 x , 6 is called the coef cient
of x . 56
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In the gure below, the perimeter of the rectangle is 20 units greater than the perimeter of each triangle. Find the length of the
diagonal of the rectangle.

Represent the perimeter of the rectangle.
P 2 l 2 w
2( 3 x 5) 2( 2 x )
Simplify this expression.

2( 3 x 5) 2( 2 x ) 2( 3 x ) 2( 5) ( 2 2) x
6 x 10 4 x
( 6 x 4 x ) 10
10 x 10
Represent the perimeter of a triangle.

P s 1 s 2 s 3
( 3 x 5) ( 2 x ) ( 4 x 10)
Simplify this expression.

( 3 x 5) ( 2 x ) ( 4 x 10) 3 x 5 2 x 4 x 10
( 3 x 2 x 4 x ) ( 5 10)
9 x 15
Write an equation.

Perimeter of rectangle Perimeter of triangle 20
10 x 10 9 x 15 20
10 x 10 9 x 5
9 x 9 x
x 10 5
10 10
x 15

3 x 5
4 x
10 2 x 57
57 Page 58 59
Objective 2
57
MAT H E
MAT I CS Use this value for x to nd the length of the diagonal of the rectangle. The diagonal is represented by the expression 4 x 10. Replace x with 15. 4 x 10 4( 15) 10 60 10
50
The diagonal of the rectangle is 50 units long. 58
58 Page 59 60
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Try It A manufacturer builds cubical containers. The equation that models
c , the cost of building a container, is c 3.6 s 3.20, in which s represents the length of each side of the cubical container in feet.

Find the length of each side of a cubical container that costs $ 35.60 to manufacture.
Substitute _ _ _ _ _ _ _ _ _ _ for c in the equation that models the cost of building a container.
Solve the equation for _ _ _ _ _ _ _ _ _ _ , the length of a side.
35.60 3.6 s 3.20

_ _ _ _ _ _ _ _ 3.6 s
32.40 3.6 s

_ _ _ _ _ _ _ _ s
Each side of the cubical container is _ _ _ _ _ _ _ _ _ _ feet long.

Substitute 35.60 for c in the equation that models the cost of building a
container. Solve the equation for s , the length of a side.

35.60 3.6 s 3.20
3.20 3.20
32.40 3.6 s
32.
3. 6
40 3.
3. 6
6s

9 s
Each side of the cubical container is 9 feet long.

Now practice what you ve learned. 59
59 Page 60 61
Objective 2
59
MAT H E
MAT I CS Question 14 Which function is the parent function of the graph below?

A y 2 x
B y x 2
C x y 2
D y x

Question 15
Marcy deposits $ 550 in a savings account at 3% simple annual interest. The value of this account,

v , is given by the function v 550 16.5 t , in
which t is the number of years the money is in the bank. What is the range of this function?

A 0 v 550
B v 550
C v 550
D 0 v 16.5

Question 16
Which of the following situations is best represented by the graph below?

A The speed of a roller coaster as it goes from a high point on its track to a low
point on the track and then back up to another high point

B The price of a stock that drops to half of its original value and then goes back to
its original value
C The speed of a bullet that is red straight up and then falls back to the ground

D The speed of a race car that starts from the starting line, races several laps, and
then makes a refueling stop

y
x
y
x
1

1
2
3
4

0 1 2 1 2

Answer Key: page 226

Answer Key: page 226 Answer Key: page 226 60
60 Page 61 62
Objective 2
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Question 17
John and some of his friends went waterskiing in his new boat. He made a table that showed the

number of gallons of gas remaining at the end of each hour. The scatterplot below shows the gas
that remained in terms of the hours that had passed.

Which of the following describes the correlation between the gas that remained and the hours
that had passed?
A Positive correlation
B No correlation
C Negative correlation
D Unde ned correlation

Question 18
Alex and Millie are selling kites from their stand on the beach. Each day more people stop to look

at and buy the kites. Alex and Millie kept the following record comparing the number of
customers that stopped to look at their kites and the money they collected in sales each day.

Kite Sales

If this trend continues, how many customers will need to stop and look at Alex and Millie s kites
in order for their sales in one day to reach $ 330?
A 42
B 60
C 66
D 48

y
x
Gas
( gallons)

Time ( hours)

Number of customers 12 18 24 30
Amount of sales ( dollars) 180 210 240 270

Answer Key: page 227 Answer Key: page 226 61
61 Page 62 63
Question 19
Frieda wants to buy a refrigerator that is 6 feet tall. The refrigerator s width is 1.75 times its

depth. Which equation best describes V , the volume of the refrigerator in terms of its depth, x ?

A V 6 x 10.5
B V x 2 1.75 x
C V 1.75 x 2 6 x
D V 10.5 x 2

Question 20
The following ordered pairs belong to the function f ( x ) .

( 1, 2) , ( 2, 8) , ( 3, 18) , ( 4, 32)
Which equation best describes this function?

A f ( x ) x 4
B f ( x ) 8 x
C f ( x ) 2 x 2
D f ( x ) x 6 x

Question 21
Which algebraic expression best represents the relationship between the terms in the following

sequence and n , their position in the sequence?
1, 3, 5, 7, 9, . . .

A n 2
B 2 n 1
C 2 n 1
D n 1

Objective 2

61
MAT H E
MAT I CS

Answer Key: page 227

Answer Key: page 227 Answer Key: page 227 62
62 Page 63 64
Question 22
Mr. Jones runs a bakery. His monthly pro t in dollars, P ( x ) , is given by the function P ( x ) 1 2 x 2, in
which x represents the number of loaves of bread sold in a month. What is the minimum number of
loaves he must sell next month if he is to have a pro t of at least $ 3000?

Record your answer and ll in the bubbles. Be sure to use the correct place value.

0
1
2
3
4
5
6
7
8
9

0
1
2
3
4
5
6
7
8
9

0
1
2
3
4
5
6
7
8
9

0
1
2
3
4
5
6
7
8
9

0
1
2
3
4
5
6
7
8
9

0
1
2
3
4
5
6
7
8
9

0
1
2
3
4
5
6
7
8
9

Objective 2

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Answer Key: page 227
Answer Key: page 228
Question 23
Which of these is equivalent to the expression below?
2 x 2( 3 x 4) 3( 8 x 4)

A 8 x 4 5
B 32 x 8
C 32 x 20
D 4( 8 x 5) 63
63 Page 64 65
The student will demonstrate an understanding of linear functions.
Objective 3

For this objective you should be able to
represent linear functions in different ways and translate among
their various representations; and

interpret the meaning of slope and intercepts of a linear function
and describe the effects of changes in slope and y -intercept in real-world and mathematical situations.

What Is a Linear Function?
A linear function is any function whose graph is a line.

MAT H E
MAT I CS

63
Is the function represented by the following table of values a linear function?
Graph the ordered pairs in the table on a coordinate grid.

The points lie on a line. Therefore, the function they represent is a linear function.
y
x
5
4

6
7
8

9
10
11
12

3
2
1
0

1
2

3
4
5
6

7
8
9
10
11

12

1 1 2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 11 12

xy
2 5
11
35
611 64
64 Page 65 66
5
3

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Try It The table below describes a linear relationship.
Does the equation 3 y 2 x 3 represent the same linear function?
To con rm that the equation 3 y 2 x 3 represents the same linear function as the table, substitute the ordered pairs from the

table into the equation.

The table and the equation represent the same function.
xy Yes/ No Does 3 y 2 x 3?
3( 1)
2( 0) 3
01 3
0 3 Yes
3 3

3 ( 5 3 )
2( 1) 3 1
5
2 3 Yes
5 5

3( 3)
2( 3) 3
33 9
6 3 Yes
9 9

xy Yes/ No Does 3 y 2 x 3?
3( _ _ _ _ _ _ )
2( _ _ _ _ _ _ ) 3
01 _ _ _ _ _ _
_ _ _ _ _ _ 3 _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _

3( _ _ _ _ _ _ )
2( _ _ _ _ _ _ ) 3
1 5 3 _ _ _ _ _ _
_ _ _ _ _ _ 3 _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _

3( _ _ _ _ _ _ )
2( _ _ _ _ _ _ ) 3
33 _ _ _ _ _ _
_ _ _ _ _ _ 3 _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _ _ _

xy
01

1 5 3
33 65
65 Page 66 67
What Is Slope?
The slope of a graph is its rate of change, how fast it increases or decreases. The slope of a line can also be described as its steepness,

how fast the line rises or falls.

The rate of change of a line is the ratio that compares the change in y -values to the corresponding change in x -values for any two points on
the graph.
A graph s slope is often described as its rise ( change in y ) over run ( change in x ) . To nd the slope of a line from its graph:

Pick any two points on the graph.
Find the change in y -values, y 2 y 1, or the rise.
Count the number of units up or down between the two points.

Find the change in x -values, x 2 x 1, or the run.
Count the number of units left to right between them.

Determine whether the slope is positive or negative.
As you go from left to right: if the line points up, the slope is positive.

if the line points down, the slope is negative.
Write the slope as a ratio:
run

rise or ch
ch
ange
ange
in
in
y
x .

y
x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 1 2 345
x = 2

Undefined slope
The line is vertical.

y

x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 1 2 345
y = 2

Zero slope
The line is horizontal.

y
x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 1 2 345
y =
x

Negative slope
The line falls from left to right.

y

x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 1 2 345
y =
x

Positive slope
The line rises from left to right.

Do you see that . . .

Objective 3

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What is the slope of the line in the graph below?
Find the slope by counting the change in y -values and the corresponding change in x -values between any two points.
The y -value changes 2 units for every 5 units the x -value changes. As you go from left to right, the line points down, so the slope is
negative.

The slope of the graph is run rise 2 5 .

y
x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 1 2 345
5 units
2 units

What is the slope of the line passing through the points ( 1, 1) and ( 3, 5) ?
Let ( x 1, y 1) be ( 1, 1) and ( x 2, y 2) be ( 3, 5) .
m x y2 2 x 1 y1 5 3 1 1 4 2 2
The slope of the line is 2.

Slope Formula
For any two points ( x 1, y 1) and ( x 2, y 2) on a graph, the slope, m , of the line that passes through them is:

m run rise x y2 2 x 1 y1 change in y -values change in x -values

Another way to nd the slope is to use the slope formula. 67
67 Page 68 69
Objective 3
67
MAT H E
MAT I CS The set of ordered pairs below represents a linear function. What is the rate of change of the function? { ( 0, 1) , ( 1, 3) , ( 2, 5) , ( 3, 7) , ( 4, 9) } Since this is a linear function, you can choose any two ordered pairs belonging to the function to nd its rate of change.
Use the ordered pairs ( 0, 1) and ( 2, 5) .
m x y2 2 x 1 y1 5 2 1 0 4 2 2
The rate of change of the function is 2.
Use the ordered pairs ( 1, 3) and ( 4, 9) .

m x y2 2 x 1 y1 9 4 3 1 6 3 2
The rate of change of the function is still 2.
The rate of change, or slope, of a linear function does not depend on the points you pick to calculate the slope. Do you see

that . . .

Try It Suppose you graphed the number of miles you drove on a trip.

Based on this graph, at what speed did you drive?
The slope of the graph represents your _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ , the
number of miles per hour you drove.
To nd the slope, compare the change in _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ and
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ between any two points on the graph.
You could use the following two points: ( 1, _ _ _ _ _ _ _ _ ) and
( _ _ _ _ _ _ _ _ , 240) .

60
120

0

180
13 2 4
Time
( h)

Distance
( mi)

y

x 68
68 Page 69 70
The y -intercept of a graph is the y -coordinate
of the point where the graph crosses the y -axis.
The x -coordinate of any point on the y -axis is 0.
Thus, the coordinates of the y -intercept are
( 0, y ) .

m 240 XX 1 XXX
_ _ _ _ _ _ _ _
Your speed was _ _ _ _ _ _ _ _ miles per hour.

The slope of the graph represents your speed, the number of miles per hour
you drove. To nd the slope, compare the change in y -values and x -values
between any two points on the graph. You could use the following two
points: ( 1, 60) and ( 4, 240) .

m 4 240 1 60

3
180

60
Your speed was 60 miles per hour.

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68
What Is the Slope-Intercept Form of a Linear Function?
One form of the equation of a linear function is y mx b . This form is called the slope-intercept form of a linear function.

When the equation of a linear function is written in the form y mx b :
m is the slope of the graph of the function; m is the value by
which the x -term is being multiplied.

b is the y -intercept of the graph of the function; b is the value
that is being added to the x -term.

What are the slope and the y -intercept of the function y 4 x 1?
The equation is in slope-intercept form, y mx b . Read the values of m and b from the equation.

The value by which the x -term is being multiplied is 4, so m 4.
The value that is being added to the x -term is 1, so b 1.

The slope of the function, m , is 4.
The y -intercept of the function, b , is 1. 69
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Objective 3
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MAT H E
MAT I CS What are the slope and y -intercept of the function 6 x y 10? To determine the slope and y -intercept, transform the equation 6 x y 10 into slope-intercept form, y mx b . 6 x y 10
6 x 6 x
y 6 x 10
The equation is now in the form y mx b .
Read the values of m and b from the revised equation. For this
function, m 6, and b 10.

The slope of the function, m , is 6.
The y -intercept of the function, b , is 10.

Try It Find the slope and y -intercept of the graph of the linear function
3 y 6 x 9.
To determine the slope and y -intercept of the graph, transform the equation 3 y 6 x 9 into slope-intercept form, y mx b .

3 y 6 x 9
3 y 6 x 9

y _ _ _ _ _ _ x _ _ _ _ _ _
Read the values of _ _ _ _ _ _ and _ _ _ _ _ _ from the revised equation.
For this function, m _ _ _ _ _ _ and b _ _ _ _ _ _ .
The slope of the function is _ _ _ _ _ _ .
The y -intercept of the function is _ _ _ _ _ _ .

3 y 6 x 9
3
3y
3
6x 9
3

y 2 x 3

Read the values of m and b from the revised equation. For this function,
m 2 and b 3. The slope of the function is 2. The y -intercept of the
function is 3. 70
70 Page 71 72
The absolute value of a number indicates its
distance from 0 on a number line. The symbol
for the absolute value of x is | x | .

For example, | 3| 3
| 5| 5
| 8.2| 8.2

Function 1 Function 2 Effect
y 1 2 xy 3 x
Since | 3| | 1 2 | , function 2 is
steeper than function 1.

y 3 x 1 y 2 x 1
Since | 2| | 3| , function 2 is less steep than function 1.

y
x
4
3
2
1

1
2
3
4

0 1 2 3 4 1 2 3 4
y = 3 x + 1
y
=

2

x
+
1

y
x
4
3
2
1

1
2
3
4

0 1 2 3 4 1 2 34
y = x
12

y
=

3
x

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What Are the Effects on the Graph of a Linear Function If the Values of m and b Are Changed in the Equation y  mx  b ?
In the equation y mx b , m represents the slope of the graph, and b represents the y -intercept of the graph. Changing either of these two
constants, m or b , will affect the graph of the function.
Change in Slope, m 71
71 Page 72 73
Objective 3
71
MAT H E
MAT I CS If the function y x 6 were changed to y 3 x 6, what would be the effect on the graph of the function? The equations are both in slope-intercept form, y mx b . In this form, m represents the graph s slope. The slope of the original function is 1.
If the equation were changed to y 3 x 6, the new value for
m would be 3.

The graph of the new function would be steeper because | 3| | 1| .
y

x
5
4

6
7
8

9

3
2
1
0

1
2

3
4
5
6

7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
y
=

3
x

6

y =
x
6

Do you see that . . . 72
72 Page 73 74
Function 1 Function 2 Effect
y 2 3 x 4 y 2 3 x 6
Function 2 crosses the y -axis at a higher point.

y x 3 y x 1
Function 2 crosses the y -axis at a lower point.

y
x
5
4

6
7
8

9

3
2
1
0

1
2

3
4
5
6

7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

y =
x
+ 3 y

=
x

1

y
x
5
4

6
7
8

9

3
2
1
0

1
2

3
4
5
6

7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
y =
x + 4 23

y =
x + 6 23

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Change in y -intercept, b
If the y -intercept of the function y 1 2 x 4 were decreased by 3, what would be the equation of the new function?
The equation is in slope-intercept form, y mx b . In this form, b represents the graph s y -intercept.
The y -intercept of the given function is 4.
If the y -intercept of the function were decreased by 3, it would be
4 3 4 ( 3) 7. The new value for b would be 7.

The equation of the new function would be y 1 2 x 7. 73
73 Page 74 75
Objective 3
73
MAT H E
MAT I CS The function y
1
4 x 1 was changed to y
1
4 x 5. What is the effect on the graph of the function?

The value for b in the rst function is 1.
Its graph crosses the y -axis at ( 0, 1) .

The value for b in the second function is 5.
Its graph crosses the y -axis at ( 0, 5) .

Since 5 1 4, the graph of y 1 4 x 5 is 4 units above the
graph of y 1 4 x 1.
y

x
5
4

6
7
8

9

3
2
1
0

1
2

3
4
5
6

7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
y = x + 1 14

y = x + 5 14

Try It If the y -intercept of the function y 2.5 x 1.75 were increased
by 2.25, what would be the effect on the graph of the function?
The equation of the given function is in slope-intercept form.
In this form, _ _ _ _ _ _ represents the slope of the line, and its
y -intercept is _ _ _ _ _ _ .
If the y -intercept were increased by 2.25, the new y -intercept would
be equal to _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .
The new line would cross the y -axis at a _ _ _ _ _ _ _ _ _ _ _ _ point than the original line.

The new line would have the same slope, _ _ _ _ _ _ , as the original line.
The equation of the given function is in slope-intercept form. In this form,
2.5 represents the slope of the line, and its y -intercept is 1.75. If the
y -intercept were increased by 2.25, the new y -intercept would be equal to
1.75 2.25 0.5. The new line would cross the y -axis at a higher point
than the original line. The new line would have the same slope, 2.5, as the
original line. 74
74 Page 75 76
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Slopes of lines can tell you whether the lines are parallel or perpendicular.
If two lines are parallel, then they have the same slope, and their
equations have the same value for m .

Look at the graphs of these two equations: y 3 x 5 and y 3 x 2.

If two lines are perpendicular, then they have negative reciprocal
slopes, and their equations have negative reciprocal values for m .

Look at the graphs of these two equations: y 4 x 1
and y 1 4 x 1.
y

x
5
4

6
7
8

9

3
2
1
0

1
2

3
4
5
6

7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
Perpendicular
lines

y
=

4
x

1

y = x 1 14

y
x
5
4

6
7
8

9

3
2
1
0

1
2

3
4
5
6

7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
y
=

3
x

5

y
=

3
x

+
2

Parallel
lines

Two numbers are negative reciprocals of each other
if their product is 1.
For example, 2 3 and 2 3
are negative reciprocals.

One fraction is the reciprocal of the other,

and one fraction is the negative of the other.

Since 2
3 2
3
6
6 1,

they are negative reciprocals. 75
75 Page 76 77
Objective 3
75
How Do You Write Linear Equations?
You can write linear equations in slope-intercept form, y mx b , or in standard form , Ax By C . In standard form, A , B , and C are

integers, and A is usually greater than zero.
You can nd the equation of a line given any of the following information:
the slope and the y -intercept of the graph

the slope and a point on the graph
two points on the graph

Given the slope and the y -intercept
Identify the values for both m , the slope, and b , the y -intercept. Write the equation in slope-intercept form, y mx b , using these values.

MAT H E
MAT I CS

What is the equation of the line with a slope of 1 3 and a y -intercept of 4?
Find the values you should substitute into the equation y mx b .
If the slope is 1
3 , then m

1
3 .
If the y -intercept is 4, then b 4.

The equation of the line is:
y mx b
y 1 3 x 4

Try It Write the equation of the line with a slope of 2 and a y -intercept of 3
5 .
If the slope is 2, then m _ _ _ _ _ _ .

If the y -intercept is 3 5 , then b .
The equation of the function is:
y mx b

y _ _ _ _ _ x

If the slope is 2, then m 2. If the y -intercept is 3 5 , then b 3 5 .
The equation of the function is:

y 2 x 3 5 76
76 Page 77 78
You can also use the point-slope form to write
the equation of a line.
y y 1 = m ( x x 1)

Given the slope and a point on the graph
Substitute the given values ( the x -and y -coordinates of the
given point and m , the slope) into the slope-intercept form of the equation, y mx b .

Solve the equation for b .
Substitute the values for m and b into the slope-intercept form,
y mx b .

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What is the equation of a line with a slope of 9, passing through the point ( 3, 4) ?
Substitute x 3, y 4, and m 9 into the slope-intercept
form of the equation.

y mx b
4 9( 3) b
Solve for b .

4 9( 3) b
4 27 b
27 27
23 b

Substitute the given value for m , 9, and the value you found
for b , 23, into the slope-intercept form of the equation.

y mx b
y 9 x 23
The equation of the line is y 9 x 23.
This equation could also be written in standard form, Ax By C , in which A is usually positive.

y 9 x 23
9 x 9 x
9 x y 23

To change 9 to a positive value, multiply both sides of the equation by 1.

9 x y 23
In standard form, the equation of this line is 9 x y 23.

Do you see that . . . 77
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Objective 3
77
MAT H E
MAT I CS Try It Write the equation of the linear graph that has a slope of 3 and contains the point ( 6, 5) . Begin by substituting the given values into the slope-intercept form.
The line passes through the point ( _ _ _ _ _ _ , _ _ _ _ _ _ ) . Therefore,
x _ _ _ _ _ _ and y _ _ _ _ _ _ .
Since the slope equals 3, m _ _ _ _ _ _ .
y mx b
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ b
Find the value for _ _ _ _ _ _ that makes the equation true.
_ _ _ _ _ _ _ _ _ _ _ _ b
b _ _ _ _ _ _
Finally, substitute the given value for m , _ _ _ _ _ _ , and the value you
found for b , _ _ _ _ _ _ , into the slope-intercept form of the equation.
y mx b
y _ _ _ _ _ x _ _ _ _ _ _
or
y _ _ _ _ _ x _ _ _ _ _ _

Begin by substituting the given values into the slope-intercept form. The line
passes through the point ( 6, 5) . Therefore, x 6 and y 5. Since the slope
equals 3, m 3.

5 3 6 b
Find the value for b that makes the equation true.
5 18 b
b 13
Finally, substitute the given value for m , 3, and the value you found for b , 13,
into the slope-intercept form of the equation.

y 3 x 13
or
y 3 x 13 78
78 Page 79 80
Given two points on the graph
Use the x -coordinates and y -coordinates of the two given points
to nd the slope, m , of the line.

Substitute the coordinates of one of the known points and the
slope you just found into the slope-intercept form of the equation, y mx b .

Solve the equation for b .
Substitute m and b into the slope-intercept form, y mx b .

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Find the equation of the line passing through the points ( 1, 2) and ( 5, 3) .
Find the slope of the graph using the slope formula. The line
passes through the points ( 1, 2) and ( 5, 3) .

m x y2 2 x 1 y1 3 5 2 1 1 4

If m is 1 4 , the slope of the line is 1 4 .
Substitute the coordinates of either of the given points and

the value you found for m , 1 4 , into the slope-intercept form of
the equation. Find the value of b .
Use ( 1, 2) or ( 5, 3) .

Substitute the values of m and b into the slope-intercept form of
the equation.

y mx b
y 1 4 x 1 3 4

The equation of the line passing through points ( 1, 2) and ( 5, 3)
is y 1 4 x 1 3 4 .

x 1, y 2
y mx b
2 1 4 ( 1) b

2 1 4 b
1
4
1
4

1 3 4 b

x 5, y 3
y mx b
3 1 4 ( 5) b

3 5 4 b
5
4
5
4

1 3 4 b 79
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Objective 3
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MAT H E
MAT I CS Try It Write the equation of the linear function that contains the points ( 4, 1) and ( 2, 3) . Find the _ _ _ _ _ _ _ _ _ _ _ _ of the graph.
The line passes through the points ( _ _ _ _ _ _ , _ _ _ _ _ _ ) and
( _ _ _ _ _ _ , _ _ _ _ _ _ ) .

m x y2 2 x 1 y1 _ _ _ _ _ _

The slope of the line is _ _ _ _ _ _ .
Substitute the coordinates of one of the given points, ( 4, _ _ _ _ _ _ ) ,
and the slope you found, _ _ _ _ _ _ , into the slope-intercept form of the
equation of a line to nd the value of b .
y mx b
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ b
_ _ _ _ _ _ _ _ _ _ _ _ b
b _ _ _ _ _ _
Substitute the values of m and b into the slope-intercept form.
y _ _ _ _ _ _ _ _ _ _ _ _

Find the slope of the graph. The line passes through the points ( 4, 1) and ( 2, 3) .
m y2 x2 y1 x1 3 2 1 4 2 2 1
The slope of the line is 1.
Substitute the coordinates of one of the given points, ( 4, 1) , and the slope
you found, 1, into the slope-intercept form of the equation of a line to nd
the value of b .

1 1 4 b
1 4 b
b 5
Substitute the values of m and b into the slope-intercept form.
y x 5 80
80 Page 81 82
How Do You Find the x -intercept and y -intercept for an Equation?
Finding the x -intercept

The x -intercept is the point where the graph of a line crosses the x -axis. The x -intercept has the coordinates ( x , 0) .

To nd the x -intercept, substitute y 0 into the equation and solve for x . The value of x is the x -intercept. The graph of the line crosses the
x -axis at the point ( x , 0) .

Finding the y -intercept
The y -intercept is the point where the graph of a line crosses the y -axis. The y -intercept has the coordinates ( 0, y ) .

One way to nd the y -intercept is to write the equation in slope-intercept form, y mx b .
The value of b is the y -intercept. The graph of the line crosses the y -axis at the point ( 0, b ) .
Another way to nd the y -intercept is to substitute x 0 into the equation and solve for y .

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Here are two ways to nd the y -intercept of the graph of y 4 2 x .
The y -intercept of the graph is 4. The graph crosses the y -axis at ( 0, 4) .
Write the equation in slope-intercept form, y mx b .
y 4 2 x 4 4
y 2 x 4
Therefore, b 4.

Substitute x 0 and solve for y .
y 4 2 x
y 4 2( 0)
y 4 0
y 4

What is the x -intercept of the graph of 5 x y 10?
Substitute y 0 into the equation and solve for x .
5 x y 10
5 x 0 10
5 x 10
x 2
The x -intercept of the graph is 2. The graph of the line crosses the x -axis at ( 2, 0) . 81
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Objective 3
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MAT H E
MAT I CS What are the x -and y -intercepts of the line of 4 x 3 y 24? To nd the y -intercept, substitute x 0 into the equation and solve for y . 4 x 3 y 24
4( 0) 3 y 24
0 3 y 24
3 y 24
y 8
The y -intercept is 8.
To nd the x -intercept, substitute y 0 into the equation and solve for x .

4 x 3 y 24
4 x 3( 0) 24
4 x 0 24
4 x 24
x 6
The x -intercept is 6. 82
82 Page 83 84
How Do You Interpret the Meaning of Slopes and Intercepts?
To interpret the meaning of the slope or of the x -or y -intercept of a function in a real-life problem, follow these guidelines:

The slope of the function is the function s rate of change. A
graph s slope tells you how fast the function s dependent variable is changing for every unit change in the independent variable.

For example, if a graph compares wages earned in dollars to hours worked, the slope tells the rate at which you are paid, or
how much you make per hour.

0
5
10
15
20
25
30
35
40

1 2 3 4 5 6 78

45
50
55
60

9 10

y
x
Hours Worked

Wages
Earned
( dollars)

y =
5.50

x

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Objective 3
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The y -intercept is the point where the graph of a function crosses
the y -axis. The y -intercept has the coordinates ( 0, y ) . It is the point in the function where the independent quantity, x , has a value of 0.

The y -intercept is often the starting point in a problem situation.
For example, if a graph describes a constant temperature drop of 10 ° F per minute from t 0 to t 8 minutes, the y -intercept

tells you what the temperature was at t 0. The y -intercept of the graph is ( 0, 85) , so the initial temperature was 85 ° F.

0
5
10
15
20
25
30
35
40

1 2 3 4 5 6 78

45
50
55
60
65
70
75
80
85

910

y
x
Time
( minutes)

Temperature
( ° F)

y = 10 x + 85 84
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Gracie took a long driving trip. She started her journey at home and drove due west for two days. The graph below represents the
number of miles Gracie was from her home on the second day of her trip. The slope of the graph is 60, and the y -intercept
is ( 0, 200) . Interpret the meaning of the slope and the y -intercept.

The slope of the graph is the function s rate of change. It compares the number of miles Gracie was from home to the number of hours
she was driving, or miles per hour. A slope of 60 means she was driving at the rate of 60 miles per hour, or that her speed was
60 mph.
The y -intercept is the point where the graph crosses the y -axis, when hours traveled equals 0. This point is the time when Gracie

started driving on the second day. A y -intercept of 200 means she was already 200 miles from home when she began driving on the
second day of her trip.

200
0

400
500

100
300

245 1 3
Hours of Driving

Distance
from Home
( miles)

y = 60 x
+ 200

y

x

The x -intercept is the point where the graph of the function crosses
the x -axis. The x -intercept has the coordinates ( x , 0) . It is the point in the problem where the dependent quantity, y , has a

value of 0 .
For example, if a graph compares the height of a falling object to the number of seconds it has fallen, the x -intercept ( when the

object s height above the ground is 0) tells you how many seconds it will take for the object to hit the ground.

0
5
10
15
20
25
30

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Time
( seconds)

Height
( feet) y = 25 16 x 2

y

x 85
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Objective 3
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MAT H E
MAT I CS Try It The graph below shows the weight of Denise s dog Elmo over the 6-month period after she adopted him. What was the dog s weight when she adopted him? How many pounds did he gain each month during that 6-month period?

The y -intercept of the graph is at the point ( 0, _ _ _ _ _ _ ) .
This means Elmo weighed _ _ _ _ _ _ pounds when Denise adopted him.

The number of pounds Elmo gained each month is the graph s rate
of _ _ _ _ _ _ _ _ _ _ _ _ , or its _ _ _ _ _ _ _ _ _ _ _ _ .
To nd the slope of the graph, identify the coordinates of _ _ _ _ _ _ points on the graph.

For example, ( 2, _ _ _ _ _ _ ) and ( _ _ _ _ _ _ , 45) are points on the graph.
The slope of the graph is the change in the _ _ _ _ _ _ -coordinates between any two points compared to the corresponding change in

their _ _ _ _ _ _ -coordinates.
The slope of the graph is _ _ _ _ _ _ .

Elmo gained _ _ _ _ _ _ pounds per month during the 6-month period.
The y -intercept of the graph is at the point ( 0, 25) . This means Elmo weighed
25 pounds when Denise adopted him. The number of pounds Elmo gained
each month is the graph s rate of change, or its slope. To nd the slope of
the graph, identify the coordinates of two points on the graph. For example,
( 2, 35) and ( 4, 45) are points on the graph. The slope of the graph is the
change in the y -coordinates between any two points compared to the
corresponding change in their x -coordinates. The slope of the graph is 45

4 2
35
2
10 5. Elmo gained 5 pounds per month during the 6-month period.

y
x
20

0

40
50

10
30

245 1 3
Months After Adoption

Weight
( pounds)

45
4 86
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Tess is lling the gas tank of her car. The graph represents the gallons of gas in her tank in terms of the number of minutes she
pumps gas.
The graph crosses the y -axis at the point ( 0, 2.5) . The graph suggests that Tess had 2.5 gallons of gasoline in her car when she

started to pump gas.
If her car had had 7 gallons of gas in it when she started pumping gas, how would the graph be affected?

If her gas tank had had 7 gallons instead of 2.5 gallons when she started pumping gas, the graph would pass through the y -axis at
the point ( 0, 7) instead of ( 0, 2.5) .
The graphs would be parallel lines.

0
5
10
15
20
25

0.5 1. 0 1.5 2. 0
Minutes

Gallons
of
Gas

y

x

How Do You Predict the Effects of Changing Slopes and y -intercepts in Applied Situations?
Many real-life problems can be modeled with linear functions. To analyze such problems, it is often helpful to identify the slope and the
y -intercept of the linear function. Interpreting the meaning of these values will help you predict the effect that changing them will have on
the quantities in the problem.
If the slope is changed, a rate of change in the problem will
increase or decrease.

If the y -intercept is changed, an initial condition will change. 87
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Objective 3
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MAT H E
MAT I CS Try It Zippy s, a package-delivery service, charges $ 12 plus $ 0.08 per mile to deliver a package within the city. How would the graph of the cost of delivering a package change if Zippy s increased its mileage charge to $ 0.09 per mile?
Write an equation that represents the cost, c , of delivering a package n miles.
c _ _ _ _ _ _ n _ _ _ _ _ _
In this equation 0.08 represents the _ _ _ _ _ _ _ _ _ _ _ _ of the graph of the equation.

If the mileage charge were changed from 0.08 to 0.09, the slope of
the graph would _ _ _ _ _ _ _ _ _ _ _ _ .
The new line will be _ _ _ _ _ _ _ _ _ _ _ _ .

Write an equation that represents the cost, c , of delivering a package
n miles.

c 0.08 n 12
In this equation 0.08 represents the slope of the graph of the equation. If the
mileage charge were changed from 0.08 to 0.09, the slope of the graph
would increase. The new line will be steeper. 88
88 Page 89 90
How Do You Solve Problems Involving Direct Variation or Proportional Change?
If a quantity y varies directly with a quantity x , then the linear equation representing the relationship between the two quantities is y kx . In
this equation, k is called the proportionality constant .
To say y varies directly with x is to say y is directly proportional to x .
If the equation y kx were graphed, k would be the slope of the graph.
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The price of milk varies directly with the number of quarts of milk purchased. If 5 quarts of milk cost $ 6.25, what would 3 quarts of
milk cost?
Write an equation that compares the number of quarts
purchased to the cost.

Let q the number of quarts purchased.
Let c the cost.
The direct variation equation is c kq .
Substitute the known values for c and q to nd the proportionality
constant, k . Five quarts of milk ( q 5) cost $ 6.25 ( c 6.25) .

c kq
6.25 k ( 5)
1.25 k
If k 1.25, then the proportionality constant is $ 1.25. If k 1.25, then the slope of the graph is 1.25.

Find the cost of 3 quarts of milk by substituting k 1.25 and
q 3 into the equation c kq .

c kq
c 1.25( 3)
c 3.75
Three quarts of milk cost $ 3.75.

Now practice what you ve learned. 89
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Objective 3
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MAT H E
MAT I CS Question 24 Which of the following can best be described by a linear function? A The area of a circle with radius r
B The perimeter of an equilateral triangle with side length s
C The surface area of a cube with side length s
D The volume of a cylinder with radius r and height h

Question 25
Which graph best represents the equation 2 y x 10?

AC

BD

y
x
5
4

6
7
8

9

3
2
1
0

1
2

3
4
5
6

7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

y
x
5
4

6
7
8

9

3
2
1
0

1
2

3
4
5
6

7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

y
x
5
4

6
7
8

9

3
2
1
0

1
2

3
4
5
6

7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

y
x
5
4

6
7
8

9

3
2
1
0

1
2

3
4
5
6

7
8
9

1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9

Answer Key: page 228

Answer Key: page 228 90
90 Page 91 92
Question 26
Which of the following linear functions does not represent a rate of change of 2 3 ?

AC

B 2 x 3 y 12 D y 2 3 x 9
Question 27
The graph below shows the number of grams of beef and the number of grams of potatoes you could eat to obtain approximately 500 calories of energy.

Which of the following numbers represents the maximum number of grams of potatoes you could eat to obtain approximately 500 calories?
A 200 g
B 450 g
C 100 g
D 150 g

0
100
200
300

100 200 300 400

400
y

Grams of Potatoes
Grams
of
Beef

x 500

x 3159
y 10 4 2 8

Objective 3

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y
x
4
3
2
1

1
2
3

0 1 2 3 1 2 34

Answer Key: page 228

Answer Key: page 228 91
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Objective 3
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MAT H E
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Answer Key: page 228
Answer Key: page 229
Answer Key: page 229

Answer Key: page 229 Answer Key: page 229

Question 28
The graph projects a business s growth in nancial assets over a seven-year period.

Which of the following interpretations of the graph is true?
A The company s initial assets are $ 200,000. The expected growth rate is $ 50 per year.
B The company s initial assets are $ 200. The expected growth rate is $ 50,000 per year.
C The company s initial assets are $ 200,000. The expected growth rate is $ 50,000 per
year.
D The company s initial assets are $ 200. The expected growth rate is $ 50 per year.

Question 29
Which of the following equations represents a graph that is parallel to the graph of the

equation 8 x 2 y 10 and that has a y -intercept of 8?

A 4 x y 5
B 4 x y 8
C 8 x 2 y 10
D 8 x 2 y 4

Question 30
The line y 3 4 x 4 is drawn on a coordinate
grid. A second line is drawn with a slope of 1.

Which statement best describes the relationship
between these two graphs?

A The second line is steeper than the rst line.
B The graphs are perpendicular lines.
C The second line is less steep than the rst line.

D The graphs are parallel lines.

Question 31
Which equation is best represented by a line containing the points ( 2, 5) and ( 4, 3) ?

A x 4 y 13
B y 4 x 13
C y 4 x 19
D 4 x y 13

Question 32
Find the coordinates of the x -intercept and the y -intercept of the line 2 x 9 3 y .

A x -intercept ( 3, 0) ; y -intercept ( 0, 9 2 )
B x -intercept ( 0, 3) ; y -intercept ( 9 2 , 0)
C x -intercept ( 0, 9 2 ) ) ; y -intercept ( 3, 0)
D x -intercept ( 9 2 , 0) ; y -intercept ( 0, 3)

0
100
200
300
400
500

1 2 3 4 5 67
Year

Assets
( thousands
of dollars)

600 92
92 Page 93 94
Question 33
A local bakery sells cookies by the dozen. If you want more than one dozen cookies, the

bakery charges by the cookie for the additional cookies. The rst graph below shows the original
cost of buying a dozen or more cookies from the bakery. The second graph shows the cost of
buying a dozen or more cookies after the bakery changed its prices.

Which statement describes how the bakery changed its price for a dozen or more cookies?
A The bakery increased the cost of the rst dozen cookies.
B The bakery increased the cost per cookie after the rst dozen.
C The bakery decreased the cost per cookie after the rst dozen.
D The bakery decreased the cost of the rst dozen cookies.

Question 34
The number of miles Sammie walks is directly proportional to the number of minutes she

walks. If Sammie walks 3 miles in 45 minutes, what is the constant of proportionality, and how
far would she walk in 2.5 hours?

A 1 1 5 ; 10 miles
B 1 1 5 ; 16.6 miles
C 15; 37.5 miles
D 15; 225 miles

0
$ 4
$ 8
$ 12
$ 16

12 16 20 24
Number of Cookies

Cost
Original
cost
New cost

Objective 3

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Answer Key: page 229 Answer Key: page 229 93
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The student will formulate and use linear equations and inequalities.
Objective 4 MAT
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93
The combined weight of 3 people in a small plane cannot safely
exceed 620 pounds. The pilot weighs 185 pounds, and Ramon
weighs 1 1 2 times as much as his wife Grace. Altogether they do
not exceed the weight limit. Write an inequality that could be used
to nd Grace s maximum possible weight.
Represent the quantities involved with variables or expressions.
You know that Ramon s weight is 1.5 times Grace s weight. Represent Grace s weight with g .

Ramon weighs 1.5 times his wife s weight, or 1.5 g .

For this objective you should be able to
formulate linear equations and inequalities from problem
situations, use a variety of methods to solve them, and analyze the solutions; and

formulate systems of linear equations from problem situations,
use a variety of methods to solve them, and analyze the solutions.

How Do You Solve Problems Using Linear Equations or Inequalities?
Many real-life problems can be solved using either a linear equation or an inequality. To solve the equation or inequality, follow these steps:

Simplify the expressions in the equation or inequality by
removing parentheses and combining like terms.

Isolate the variable as a single term on one side of the equation
by adding or subtracting expressions on both sides of the equation or inequality.

Use multiplication or division to produce a coef cient of 1 for
the variable term.

When solving an inequality, you must reverse the inequality
symbol if you multiply or divide both sides by a negative number.

2 x 10

2
2x 10
2 Divide both sides by a negative number.
x 5

The inequality symbol reversed; it went from to .
Use the solution of the equation or inequality to nd the answer
to the question asked.

See whether your answer is reasonable.

An algebraic sentence written using the symbol
, , , or is called an inequality.

Do you see that . . . 94
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Write the inequality.
Grace s weight Ramon s weight the pilot s weight must be less than or equal to 620 pounds.

g 1.5 g 185 620
1 g 1.5 g 185 620
2.5 g 185 620
The inequality 2.5 g 185 620 could be used to nd Grace s maximum possible weight.

In the school Adopt-a-Highway program, Trent picked up twice as many empty soda cans as Susan did but only one-third as many as
Ginger collected. Together, the team picked up 135 cans. How many cans did Trent pick up?

Represent the unknown quantities with variables or expressions.
The number of cans each team member picked up is described in terms of the number of cans Trent picked up. Represent the

number of cans Trent picked up using the variable t .
Susan picked up 1 2 as many cans as Trent.
Ginger picked up 3 times as many cans as Trent.
Trent Susan Ginger
t 0.5 t 3 t
Write an equation that can be used to solve the problem.
Together the team picked up 135 cans.

t 0.5 t 3 t 135
Simplify the expression by combining like terms. Then solve
the equation for t .

1 t 0.5 t 3 t 135
( 1 0.5 3) t 135
4.5 t 135
t 30
In this problem, the solution to the equation, t 30, is also the answer to the problem. Trent picked up 30 empty soda cans. 95
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Objective 4
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MAT H E
MAT I CS The width of a rectangle is between 5 and 10 inches. If its length is 1 inch more than twice its width, what is a reasonable range for the rectangle s perimeter? Its width can be represented by w . Its length can be represented by 2 w 1.
Its perimeter can be represented by P .
P 2 l 2 w
P 2( 2 w 1) 2 w
P 4 w 2 2 w
P 6 w 2
Find a reasonable range for P .
Substitute Substitute w 5 w 1 0

( 6 w 2) P ( 6 w 2)
( 6 5 2) P ( 6 10 2)
( 30 2) P ( 60 2)
32 P 62
Any perimeter between 32 inches and 62 inches would be reasonable.

An illustrator wants to produce a series of pen-and-ink drawings. The height of each drawing must be at least 1 inch more than
1
3 its width. This relationship is represented in the graph below.

If a drawing 6 inches wide is 5 inches tall, does it meet the requirements?
Find the 6 on the x -axis and then go up to the shaded region in the graph. Is the point ( 6, 5) in the shaded region? Yes.
Then a 6-inch-by-5-inch drawing meets the requirements.

0
2
4
6

2 4 6810
Height
( inches)

Width
( inches)

Reasonable solutions
lie in the shaded area.
y

x
( 6, 5) 96
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Try It The length of a rectangle is 5 inches greater than its width. If the
width of the rectangle is represented by w and its perimeter is represented by P , write an equation in terms of P and w that could
be used to nd the dimensions of the rectangle.
Represent the width of the rectangle by w .

The length of the rectangle is 5 inches _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ its width.

Represent the length by w _ _ _ _ _ _ _ _ _ _ . Write an equation and simplify it.
P 2 l 2 w
P 2( w _ _ _ _ _ _ _ _ _ _ ) 2 w
P 2 w _ _ _ _ _ _ _ _ _ _ 2 w
P _ _ _ _ _ _ _ _ _ _ w _ _ _ _ _ _ _ _ _ _
The equation P _ _ _ _ _ _ _ _ _ _ w _ _ _ _ _ _ _ _ _ _ could be used to nd the dimensions of the rectangle.

The length of the rectangle is 5 inches greater than its width. Represent the
length by w 5.

P 2 l 2 w
P 2( w 5) 2 w
P 2 w 10 2 w
P 4 w 10
The equation P 4 w 10 could be used to nd the dimensions of the
rectangle. 97
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Objective 4
97
MAT H E
MAT I CS Try It Saul used two different types of trim molding on a woodworking project. He used 6 feet more of the molding that cost $ 2.50 per foot than he used of the molding that cost $ 1.50 per foot. If the combined cost of the molding used to complete his project was $ 55, how many
feet of each type of molding did Saul use?
Represent the number of feet of $ 1.50-per-foot molding with m .
Represent the number of feet of $ 2.50-per-foot molding with
m _ _ _ _ _ _ _ _ _ _ .

Represent the cost of the $ 1.50 molding used.
_ _ _ _ _ _ _ _ _ _
Represent the cost of the $ 2.50 molding used.
2.50( _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ )
Write an equation that shows the total cost as $ 55 and solve for m .
1.50 m 2.50( _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ) 55
1.50 m 2.50 m _ _ _ _ _ _ _ _ _ _ 55
_ _ _ _ _ _ _ _ _ _ m 15 55
_ _ _ _ _ _ _ _ _ _ m _ _ _ _ _ _ _ _ _ _
m _ _ _ _ _ _ _ _ _ _
Saul used _ _ _ _ _ _ _ _ _ _ feet of molding that cost $ 1.50 per foot.
He used m 6 _ _ _ _ _ _ _ _ _ _ 6 _ _ _ _ _ _ _ _ _ _ feet of molding that cost $ 2.50 per foot.

Represent the number of feet of $ 2.50-per-foot molding with m 6.
Represent the cost of the $ 1.50 molding used: 1.50 m . Represent the cost
of the $ 2.50 molding used: 2.50( m 6) .

1.50 m 2.50( m 6) 55
1.50 m 2.50 m 15 55
4 m 15 55
4 m 40
m 10
Saul used 10 feet of molding that cost $ 1.50 per foot. He used
m 6 10 6 16 feet of molding that cost $ 2.50 per foot. 98
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98
Try It A farmer sells peaches and decorative baskets at a roadside stand.
He sells the baskets for $ 4.95 each and the peaches for $ 0.69 per pound. The equation p 0.69 w 4.95 represents the price, p ,
of a basket containing w pounds of peaches. The graph of this relationship is shown below.

Use the graph to nd a reasonable value for the number of pounds of peaches in a decorative basket that would sell for $ 10.00 altogether.
Go to the point on the _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ axis where the cost is approximately $ 10.00.
Go _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ until you reach the graph.
Go _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ to the _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ axis and read the value there.

The corresponding value is greater than _ _ _ _ _ _ _ _ _ _ .
A reasonable value for the number of pounds of peaches in a basket
costing $ 10.00 is about _ _ _ _ _ _ _ _ _ _ pounds.

Go to the point on the vertical axis where the cost is approximately $ 10.00.
Go across until you reach the graph. Go down to the horizontal axis and read
the value there. The corresponding value is greater than 7. A reasonable value
for the number of pounds of peaches in a basket costing $ 10.00 is

about 7 1 3 pounds.

16
10
15
11
12
13
14

0
1
2
3
4
5
6

1 2 3 4 5 6 7 891011

7
8
9

12
Weight
( pounds)

Price
( dollars)

p

w 99
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Objective 4
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MAT H E
MAT I CS
How Do You Represent Problems Using a System of Linear Equations?
Many real-life problems can be solved using a system of two or more linear equations.

To r epresent a problem using a system of linear equations, follow these guidelines.
Identify the quantities involved and the relationships between
them.

Represent the quantities involved with two different variables or
with expressions involving two variables.

Write two independent equations that can be used to solve the
problem.

The sum of two numbers is 100. Twice the rst number plus three times the second number is 275. Write a system of two equations
in two unknowns that could be used to nd the two numbers.
The problem involves two numbers. One number is not described in terms of the other number, so it makes sense to represent them

using two different variables.
Represent the rst number with the variable x .

Represent the second number with the variable y .

You know two different relationships between the numbers.
Their sum is 100.

x y 100
Twice the rst number plus three times the second number is 275.

twice the rst three times the second 275
2 x 3 y 275
The following system of linear equations could be used to nd the two numbers.

x y 100
2 x 3 y 275

A system of linear equations is two or more
linear equations that use two or more variables.
For example,
2 x y 10
x 3 y 9

is a system of two linear equations in

two unknowns. 100
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100
Try It Daniel put all the pennies and nickels from his pocket change into a
jar. At the end of the month, there were 210 coins in his jar, worth $ 3.30 in all. Write a system of two equations that can be used to
nd p , the number of pennies, and n , the number of nickels, in the jar at the end of the month.

The value of a group of mixed coins depends on the value of each type of coin.

The number of any one type of coin times its _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ gives the total value of all the coins of that type.
Represent the number of coins of each type Daniel had in his jar. Represent the number of pennies with the variable p .
Represent the number of nickels with the variable n .
Represent the total value in cents of each type of coin in his jar.

Represent the value of the pennies with the expression _ _ _ _ _ _ _ _ _ _ .
Represent the value of the nickels with the expression _ _ _ _ _ _ _ _ _ _ .
Write two equations that describe the relationships between these quantities:

The number of pennies the number of nickels _ _ _ _ _ _ _ _ _ _ coins
in the jar:
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
The value of the pennies the value of the nickels _ _ _ _ _ _ _ _ _ _
cents in the jar:
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
The following system of equations could be used to nd the number of each type of coin Daniel has in his jar:

p n _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ _ _ p _ _ _ _ _ _ _ _ n _ _ _ _ _ _ _ _ _ _ 101
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Objective 4
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The solution to a system of linear equations is a pair
of numbers that makes both equations true.

For example, the ordered pair ( 4, 2) is a solution
to the following system of equations:

2 x y 10
x y 2

because when x 4 and y 2, each equation is

true.

The number of any one type of coin times its value gives the total value
of all the coins of that type. Represent the value of the pennies with the
expression 1 p . Represent the value of the nickels with the expression 5 n .
The number of pennies the number of nickels 210 coins in the jar:
p n 210. The value of the pennies the value of the nickels 330
cents in the jar: 1 p 5 n 330. The following system of equations could
be used to nd the number of each type of coin Daniel has in his jar:

p n 210
1 p 5 n 330

If the two lines as shown by this graph then the system of equations
intersect at a has one solution: single point x 1 and y 2
or ( 1, 2) .

do not intersect ( parallel lines) has no solution.

intersect at every point has many solutions.
( the same line)

y
x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 12345
2 x
+
y
=
4

4 x
+
2 y

=
8

y
x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 12345
2 x
+
y
=
4

2 x
+
y
=
1

How Do You Solve a System of Linear Equations?
You can solve a system of linear equations algebraically and graphically. Two algebraic methods are substitution and elimination.

A graph can show you how many solutions a system of equations has.

y
x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 12345
2 x
+
y
=
4

x
y =
1

( 1, 2)

MAT H E
MAT I CS 102
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You can also determine the number of solutions a system of equations has by writing the equations in slope-intercept form, y mx b .
Solve this system of equations using the graphical method.
x y 2
2 x y 5
The two equations are graphed below.

The coordinates of the point where the two lines intersect, ( 3, 1) , is the solution to the system of equations. The coordinates satisfy
both equations.

The solution to the system of equations is ( 3, 1) .

y
x
5
4
3
2
1

1
2
3
4
5

0 1 2 3 4 5 1 2 345
x
+ y

=
2

2 x

y
=

5

( 3, 1)

x y 22 x y 5
x 3 x y 22 x y 5
and ( 3) ( 1) 2 2( 3) ( 1) 5 y 1 2 25 5

If the two equations as shown by this system then the system of equations
have different values 2 x y 4 y 2 x 4has one solution: for mx y 1 y x 1 x 1 and y 2.
have different values for b but the same 2 x y 4 y 2 x 4 ( b 4) has no solution.
value for m 2 x y 1 y 2 x 1 ( b 1)
have the same 2 x y 4 y 2 x 4 has many solutions. values for m and b 4 x 2 y 8 y 2 x 4 103
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Objective 4
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MAT H E
MAT I CS Solve the following system of equations using the substitution
method.
a b 30
2 a 3 b 10
This system of equations lends itself to being solved using substitution because the rst equation can be solved easily for

a in terms of b .
To solve the rst equation for a , subtract b from both sides.
a b 30
b b
a 30 b
Now substitute ( 30 b ) for a in the second equation.
The result will be a linear equation with just one variable.
Solve that equation.
2 a 3 b 10
2( 30 b ) 3 b 10 Substitute ( 30 b ) for a .
60 2 b 3 b 10 Remove parentheses; multiply by 2.
60 5 b 10 Combine like terms: 2 b 3 b 5 b .
60 10 5 b Add 5 b to both sides.
50 5 b Subtract 10 from both sides.
10 b Divide both sides by 5.
Now that you know the value of b , you can use this value to nd the value of a by substituting the value of b into either of the two

original equations.
In this system, the rst equation is the easier equation to use.
a b 30
a 10 30 Substitute b 10.
a 20 Subtract 10 from both sides.
The solution to the system of equations is a 20 and b 10, or ( 20, 10) .

When you use the substitution method to solve a system of equations, solve one equation for one of the two variables. Then substitute into
the second equation. 104
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Solve the same system of equations using the elimination method.
a b 30
2 a 3 b 10
In the given system of equations, if you multiply both sides of the rst equation by 3, you get the following equations:

3 ( a b) 3( 30) 3 a 3 b 90
2 a 3 b 10 2 a 3 b 10
The variable b has a coef cient of 3 in the new equation and an opposite coef cient of 3 in the original second equation. When

you add those two equations, the term containing b will disappear because it will have a 0 coef cient.

3 a 3 b 90
2 a 3 b 10
5 a 100
a 20
To nd b , substitute 20 for a into the rst equation.
a b 30
20 b 30
b 10
The solution to the system of equations is a 20 and b 10, or ( 20, 10) . This is the same solution obtained by using the

substitution method.

Do you see that . . .

The elimination method is also called the addition method because you eliminate one of the variables by adding. Before you add, you may need
to multiply one or both equations so that one of the variables has opposite coef cients. 105
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Objective 4
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MAT H E
MAT I CS Try It At the concession stand a can of soda costs $ 0.25 more than a bottle of water. If John bought 3 bottles of water and 2 cans of soda for $ 8, how much did each type of drink cost? Let s represent the cost of a can of soda.
Let w represent the cost of a bottle of water.
If a can of soda costs $ 0.25 more than a bottle of water, then an equation that can be used to represent this is

s w _ _ _ _ _ _ _ _ _ _ .
If 3 bottles of water and 2 cans of soda cost $ 8, then an equation that can be used to represent this is

3_ _ _ _ _ _ _ _ _ _ 2_ _ _ _ _ _ _ _ _ _ 8.
Solve the system of equations using the substitution method. Substitute w 0.25 for s in the second equation and solve.

3 w 2 s 8.00
3 w 2( _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ) 8.00
3 w _ _ _ _ _ _ _ _ _ _ w _ _ _ _ _ _ _ _ _ _ 8.00
_ _ _ _ _ _ _ _ _ _ w 0.50 8.00
_ _ _ _ _ _ _ _ _ _ w 7.50
w _ _ _ _ _ _ _ _ _ _
A bottle of water costs $ _ _ _ _ _ _ _ _ _ _ .
A can of soda costs s w 0.25 _ _ _ _ _ _ _ _ _ 0.25 $ _ _ _ _ _ _ _ _ _ .

If a can of soda costs $ 0.25 more than a bottle of water, then an equation
that can be used to represent this is s w 0.25. If 3 bottles of water and
2 cans of soda cost $ 8, then an equation that can be used to represent this
is 3 w 2 s 8.

3 w 2 s 8.00
3 w 2( w 0.25) 8.00
3 w 2 w 0.50 8.00
5 w 0.50 8.00
5 w 7.50
w 1.50
A bottle of water costs $ 1.50.
A can of soda costs s w 0.25 1.50 0.25 $ 1.75.

Now practice what you ve learned. 106
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Question 35
Jeanne wants to build a fence to enclose an area for a new rose garden. She can afford 150 feet of

fencing. The length of the rectangular garden will be 5 feet more than the width, w . Which inequality
best describes the possible width of her garden?
A w ( w 5) 150
B 2 w 2( w 5) 150
C 2 w 2( w 5) 150
D w ( w 5) 150

Question 36
Sharon kept track of her expenses last week. She spent $ 4 more on movie rentals than she did on

lunches. She spent ve times as much xing her car as she did on movie rentals. If Sharon spent
a total of $ 80 last week on these three expenses, how much did her car repairs cost?

A $ 8
B $ 12
C $ 60
D $ 14

Question 37
The sum of two numbers is 59. The difference between 2 times the rst number and 6 times

the second is 34. Find the two numbers.
A 40 and 19
B 38 and 97
C 38 and 97
D 40 and 19

Question 38
Each morning at his bagel shop, Sid makes at least three times as many plain bagels as he does

onion bagels and 25 more onion bagels than garlic bagels. The graph below represents the
relationship between the number of plain bagels, p , and the number of garlic bagels, g , Sid
prepares each day.

Which statement below does not satisfy this inequality relationship?
A Sid made 100 garlic bagels and 390 plain bagels.
B Sid made 50 garlic bagels and 225 plain bagels.
C Sid made 25 garlic bagels and 125 plain bagels.
D Sid made 75 garlic bagels and 300 plain bagels.

275
300
325
350
375
400

225
250