Algebra 1 G9

Get Ready! CHAPTER

Lesson 1-9 Solutions of a Two-Variable Equation 5

Tell whether the given ordered pair is a solution of the equation.

1. 4y + 2x = 3; (1.5, 0) 2. y = 7x - 5; (0, 5) 3. y = -2x + 5; (2, 1)

Lesson 2-5 Transforming Equations

Solve each equation for y.

4. 2y - x = 4 5. 3x = y + 2 6. -2y - 2x = 4

Lesson 2-6 Comparing Unit Rates

7. Transportation  A car traveled 360 km in 6 h. A train traveled 400 km in 8 h. A boat
traveled 375 km in 5 h. Which had the fastest average speed?

8. Plants  A birch tree grew 2.5 in. in 5 months. A bean plant grew 8 in. in 10 months.
A rose bush grew 5 in. in 8 months. Which grew the fastest?

Lesson 4-4 Graphing a Function Rule

Make a table of values for each function rule. Then graph each function.

9. f (x) = x + 3 10. f (x) = -2x 11. f (x) = x - 4

Lesson 4-7 Arithmetic Sequences

Write an explicit formula for each arithmetic sequence.

12. 2, 5, 8, 11, c 13. 13, 10, 7, 4, c 14. -3, -0.5, 2, 4.5, c

Looking Ahead Vocabulary

15. A steep hill has a greater slope than a flat plain. What does the slope of a line on a
graph describe?

16. Two streets are parallel when they go the same way and do not cross. What does it
mean in math to call two lines parallel?

17. John was bringing a message to the principal’s office when the principal
intercepted him and took the message. When a graph passes through the y-axis, it
has a y-intercept. What do you think a y-intercept of a graph represents?

Chapter 5  Linear Functions 291

5CHAPTER Linear Functions

Download videos VIDEO Chapter Preview 1 Proportionality
connecting math Essential Question:  What does the slope
to your world.. 5-1 Rate of Change and Slope of a line indicate about the line?
5-2 Direct Variation
Interactive! ICYNAM 5-3 Slope-Intercept Form 2 Functions
Vary numbers, ACT I V I TI 5-4 Point-Slope Form Essential Question:  What information
graphs, and figures D 5-5 Standard Form does the equation of a line give you?
to explore math ES 5-6 Parallel and Perpendicular Lines
concepts.. 5-7 Scatter Plots and Trend Lines 3 Modeling
5-8 Graphing Absolute Value Functions Essential Question:  How can you make
predictions based on a scatter plot?

The online
Solve It will get
you in gear for
each lesson.

Math definitions VOC ABUL ARY Vocabulary DOMAINS
in English and • Interpreting Functions
Spanish English/Spanish Vocabulary Audio Online: • Building Functions
• I nterpreting Categorical and Quantitative Data
English Spanish

direct variation, p. 301 variación directa

Online access linear equation, p. 308 ecuación lineal
to stepped-out
problems aligned piecewise function, p. 348 función de fragmentos
to Common Core
Get and view point-slope form, p. 315 forma punto-pendiente
your assignments
online. NLINE rate of change, p. 294 tasa de cambio
ME WO
O slope, p. 295 pendiente
RK
HO slope-intercept form, p. 308 forma pendiente-intercepto

standard form, p. 322 forma normal

step function, p. 348 función escalón

Extra practice trend line, p. 337 línea de tendencia
and review
online x-intercept, p. 322 intercepto en x

y-intercept, p. 308 intercepto en y

Virtual NerdTM
tutorials with
built-in support

PERFORMANCE TASK

Common Core Performance Task

Marbles in Water

Have you noticed that dropping objects into a glass of water raises the water level?
The diagram and table below show what happens to the height of the water in a
certain glass when different numbers of identical marbles are dropped into it.

15 cm Number of Height of
Marbles, x Water (cm), y

3 6.9
7.5
5 8.7
10.2
9 11.1
12.9
6.9 cm 14
17

23

  

Task Description

Predict the number of marbles you need to drop in the glass to raise the water
level to the top of the glass.

Connecting the Task to the Math Practices MATHEMATICAL

As you complete the task, you’ll apply several Standards for Mathematical PRACTICES
Practice.

• You’ll analyze the data in the table to determine the type of relationship
between the number of marbles in the glass and the water height. (MP 1)

• You’ll model the data in the table with a function. (MP 2, MP 4)

Chapter 5  Linear Functions 293

5-1 Rate of Change CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
and Slope
MF-LAEF.SA..911b2 .FR-eLcEo.g1n.i1zbe  sRiteucaotigonniszeinsiwtuhaictihonosneinqwuahnicthity
Objectives To find rates of change from tables ochnaenqgueasnattitay choannstgaenstartataecpoenrsutannit irnatteervpaelrruenlaitivinetetorval
To find slope arenloatihveer.tAo lasnooFth-eIFr.BA.l6so MAFS.912.F-IF.2.6

MP 1, MP 2, MP 3, MP 4

The table shows the horizontal and vertical Horizontal Vertical
distances from the base of the mountain at Pole Distance Distance
several poles along the path of a ski lift. The
poles are connected by cable. Between which A 20 30
two poles is the cable’s path the steepest?
How do you know? B 40 35

Drawing a diagram C 60 60
may help.
D 100 70
MATHEMATICAL

PRACTICES

Lesson Essential Understanding  You can use ratios to show a relationship between
changing quantities, such as vertical and horizontal change.
Vocabulary
• rate of change Rate of change shows the relationship between two changing quantities. When one
• slope quantity depends on the other, the following is true.

rate of change = change in the dependent variable
change in the independent variable

Problem 1 Finding Rate of Change Using a Table

Does this problem Marching Band  The table shows the distance a band marches Distance Marched
look like one you’ve over time. Is the rate of change in distance with respect to
seen before? time constant? What does the rate of change represent? Time Distance
Yes. In Lesson 2-6, you (min) (ft)
wrote rates and unit
rates. The rate of change rate of change = change in distance 1 260
in Problem 1 is an change in time
example of a unit rate.
Calculate the rate of change from one row of the table to the next. 2 520

520 - 260 = 2610 780 - 520 = 2610 1040 - 780 = 260 3 780
2 - 1 3 - 2 4-3 1
4 1040
The rate of change is constant and equals 126m0ifnt . It represents
the distance the band marches per minute.

Got It? 1. In Problem 1, do you get the same rate of change if you use
nonconsecutive rows of the table? Explain.

294 Chapter 5  Linear Functions

The graphs of the ordered pairs (time, distance) in Problem 1 Distance Marched
lie on a line, as shown at the right. The relationship between
time and distance is linear. When data are linear, the rate of Distance (ft) 1000 up
change is constant. 600 260 units
200 right
Notice also that the rate of change found in Problem 1 is 1 unit
just the ratio of the vertical change (or rise) to the horizontal 00
change (or run) between two points on the line. The rate of 12 34
change is called the slope of the line. Time (min)

slope = vertical change = rise
horizontal change run

Problem 2 Finding Slope Using a Graph

What do you need to What is the slope of each line? B 4y
f ind the slope? (؊3, 2)
You need to find the rise A right 3 4 y down
and run. You can use the units 4 units 2
graph to count units of
rise and units of run. O (1, 1) x Ϫ4 Ϫ2 Ox
24 2(2, ؊42)
up (؊2, ؊1)
2 units Ϫ4

right
5 units

slope = rise slope = rise
run run

= 2 = -4 = - 4
3 5 5

The slope of the line is 2 . The slope of the line is - 4 .
3 5

Got It? 2. What is the slope of each line in parts (a) and (b)? 2y
a. 3 y b.

1 x x
Ϫ4 Ϫ2 O 24
Ϫ4 Ϫ2 O 2
Ϫ2

c. Reasoning  In part (A) of Problem 2, pick two new points on the line to
find the slope. Do you get the same slope?

Notice that the line in part (A) of Problem 2 has a positive slope and slants upward from
left to right. The line in part (B) of Problem 2 has a negative slope and slopes downward
from left to right.

Lesson 5-1  Rate of Change and Slope 295

You can use any two points on a line to find its slope. Use subscripts x2 ؊ x1
B(x2, y2)
to distinguish between the two points. In the diagram, (x1, y1) are y2 ؊ y1
the coordinates tohfepsolionpteAo,fan·AdB(x, 2y,oyu2)caarneuthsee coordinates of
point B. To find the slope formula. A(x1, y1)

Key Concept  The Slope Formula

slope = rise = y2 - y1 , where x2 - x1 ≠ 0
run x2 - x1

The x-coordinate you use first in the denominator must belong to the same ordered

pair as the y-coordinate you use first in the numerator.

Problem 3 Finding Slope Using Points
What is the slope of the line through (−1, 0) and (3, −2)?

Does it matter which You need the slope, so start slope = y2 − y1 –– 1 / 2
point is (x1, y1) and with the slope formula. x2 − x1
which is (x2, y2)? 00000 0
No. You can pick either Substitute ( - 1, 0) for (x1, y1) = −2 −0 11111 1
point for (x1, y1) in the and (3, - 2) for (x2, y2). 3− ( − 1) 22222 2
slope formula. The other 33333 3
Simplify to find the answer −2 44444 4
point is then (x2, y2). to place on the grid. 4 55555 5
66666 6
77777 7
88888 8
99999 9

= = −21

Got It? 3. a. What is the slope of the line through (1, 3) and (4, -1)?
b. R easoning  Plot the points in part (a) and draw a line through them. Does

the slope of the line look as you expected it to? Explain.

Problem 4 Finding Slopes of Horizontal and Vertical Lines

What is the slope of each line?

A (؊3, 2) 3 y (2, 2) B 2y
Can you generalize (؊2, 1) x

these results? Ϫ2 O x Ϫ4 O
Yes. All points on a 2 (؊2, ؊2)
horizontal line have the Ϫ2

same y-value, so the

Fstsvollieono rdpdptiieiecvn aiigissslioltaaihnnllwewebasaaylyyo lwzsspeuaezrnoyeos.drf oTel he.afiaendesd. LTslheotep(sexl1o=,pyxye122)o--=f tyxh(11-e=3h,o22r2-)iza-(on-n2d3ta)(lx=l2i,n50ye=2i)s0=0 .( 2, 2). Let (x1, y1) = ( - 2, - 2) and (x2, y2) = ( - 2, 1).
y2 - y1 1 - ( - 2) 3
slope = x2 - x1 = - 2 - ( - 2) = 0

D ivision by zero is undefined. The

slope of the vertical line is undefined.

296 Chapter 5  Linear Functions

Got It? 4. What is the slope of the line through the given points?
a. (4, -3), (4, 2) b. ( -1, -3), (5, -3)

The following summarizes what you have learned about slope.

Concept Summary  Slopes of Lines

A line with positive slope y A line with negative slope y
slants upward from left slants downward from left Ox
to right. x to right.
O

A line with a slope of 0 y A line with an undefined y
is horizontal. slope is vertical.
x x
O O

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
1. Is the rate of change in cost constant with respect to
4. Vocabulary  What characteristic of a graph represents
the number of pencils bought? Explain.
the rate of change? Explain.

Cost of Pencils 5. Open-Ended  Give an example of a real-world
situation that you can model with a horizontal line.
Number of Pencils 1 4 7 12 What is the rate of change for the situation? Explain.
1.75 3
Cost ($) 0.25 1 6. Compare and Contrast  How does finding a line’s
slope by counting units of vertical and horizontal
2. What is the slope of the line? change on a graph compare with finding it using the
3y slope formula?

Ϫ2 O 2x 7. Error Analysis  A student y
Ϫ2 calculated the slope of the
line at the right to be 2. 3 up
Explain the mistake. What is right1 unit
the correct slope? 2 units x

Ϫ2 O 2 4

3. What is the slope of the line through ( -1, 2)
and (2, -3)?

Lesson 5-1  Rate of Change and Slope 297

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Determine whether each rate of change is constant. If it is, find the rate of See Problem 1.
change and explain what it represents.

8. Turtle Walking 9. Hot Dogs and Buns 10. Airplane Descent
Time (min) Distance (m) Time (min) Elevation (ft)
16 Hot Dogs Buns 0 30,000
2 12 1 1 2 29,000
3 15 5 27,500
4 21 22 12 24,000

33

44

Find the slope of each line. See Problem 2.

11. y 12. y 13. y

2 2 2 x

x Ϫ2 O 3x Ϫ2 O 2
Ϫ2 O 2 Ϫ2 Ϫ2

Ϫ2

14. 6 y 15. 4y x 16. 4 y x
4 Ϫ4 2 4 2 4
2
O2 Ϫ4 Ϫ2 O 2
x Ϫ2
Ϫ2 O 2 4 6 Ϫ4
Ϫ2
Ϫ4

Find the slope of the line that passes through each pair of points. See Problem 3.
19. (4, 4), (5, 3)
17. (0, 0), (3, 3) 18. (1, 3), (5, 5) 22. (2, -3), (5, -4)

20. (0, -1), (2, 3) 21. ( -6, 1), (4, 8)

Find the slope of each line. See Problem 4.
23. 4 y
24. y 25. y

Ϫ4 Ϫ2 O x 2 x 2 x
Ϫ2 24 Ϫ4 Ϫ2 O 24 24
Ϫ4 Ϫ2 O
Ϫ2

Ϫ4

298 Chapter 5  Linear Functions

B Apply Without graphing, tell whether the slope of a line that models each linear
relationship is positive, negative, zero, or undefined. Then find the slope.

26. The length of a bus route is 4 mi long on the sixth day and 4 mi long on the
seventeenth day.

27. A babysitter earns $9 for 1 h and $36 for 4 h.

28. A student earns a 98 on a test for answering one question incorrectly and earns a 90
for answering five questions incorrectly.

29. The total cost, including shipping, for ordering five uniforms is $66. The total cost,
including shipping, for ordering nine uniforms is $114.

State the independent variable and the dependent variable in each linear
relationship. Then find the rate of change for each situation.

30. Snow is 0.02 m deep after 1 h and 0.06 m deep after 3 h.

31. The cost of tickets is $36 for three people and $84 for seven people.

32. A car is 200 km from its destination after 1 h and 80 km from its destination after 3 h.

Use the slope formula to find the slope of the line that passes through each pair of
points. Then plot the points and sketch the line that passes through them. Does the
slope you found using the formula match the direction of the line you sketched?

33. ( -2, 1), (7, 1) 34. (4.25, 0), (3.5, 3)

( ) ( ) 35. -21, 47 , 8, 74 36. ( -5, 0.124), ( -5, -0.584)

37. ( -42.25, 5.2), (3.25, 3) -( ) ( )38. 2,2 , - 2, 7
11 13

39. Think About a Plan  The graph shows the average growth rates Rate of Growth
for three different animals. Which animal’s growth shows the
fastest rate of change? The slowest rate of change? Weight (kg) horse seal
30
• How can you use the graph to find the rates of change?
• Are your answers reasonable? 15

40. Open-Ended  Find two points that lie on a line with slope -9. 00 mouse
20 40 60 80 100
41. Profit  John’s business made $4500 in January and $8600
in March. What is the rate of change in his profit for this Days
time period?

Each pair of points lies on a line with the given slope. Find x or y.

42. (2, 4), (x, 8); slope = -2 43. (4, 3), (5, y); slope = 3
44. (2, 4), (x, 8); slope = - 12
46. ( -4, y), (2, 4y); slope = 6 45. (3, y), (1, 9); slope = - 5
2

47. (3, 5), (x, 2); undefined slope

48. Reasoning  Is it true that a line with slope 1 always passes through the origin?
Explain your reasoning.

Lesson 5-1  Rate of Change and Slope 299

49. Arithmetic Sequences  Use the arithmetic sequence 10, 15, 20, 25, c
a. Find the common difference of the sequence.
b. Let x = the term number, and let y = the corresponding term of the sequence.

Graph the ordered pairs (x, y) for the first eight terms of the sequence. Draw a

line through the points.
c. Reasoning  How is the slope of a line from part (b) related to the common

difference of the sequence?

C Challenge Do the points in each set lie on the same line? Explain your answer.

50. A(1, 3), B(4, 2), C( -2, 4) 51. G(3, 5), H( -1, 3), I(7, 7) 52. D( -2, 3), E(0, -1), F(2, 1)
53. P(4, 2), Q( -3, 2), R(2, 5) 54. G(1, -2), H( -1, -5), I(5, 4) 55. S( -3, 4), T(0, 2), X( -3, 0)

Find the slope of the line that passes through each pair of points.

56. (a, -b), ( -a, -b) 57. ( -m, n), (3m, -n) 58. (2a, b), (c, 2d)

PERFORMANCE TASK MATHEMATICAL

Apply What You’ve Learned PRACTICES
MP 1

The table at the right shows the height of water in a glass when different
numbers of marbles are dropped into it.

a. Find the rate of change in the water height with respect to the Number of Height of
number of marbles from one row in the table to the next. Marbles, x Water (cm), y
What do you notice?
3 6.9

b. For each marble you add to the glass, how much does the 5 7.5

water level rise? Number of 9 8.7
mModaerblsles, x WaHteeirg1h(4ctmo)f, y 10.2
c. What can you conclude about the type of function that 11.1
167.9 12.9
the relationship between the number of marbles and the water3 273.5
height? Explain.

5

9 8.7

14 10.2

17 11.1

23 12.9

300 Chapter 5  Linear Functions

5-2 Direct Variation CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

MA-ACFESD..9A1.2. AC-rCeEatDe.1eq.2u aCtiroenasteineqtwuaotoiornms oinretwvaorioarbmleosre
vtoarrieapbrlesetnotrreeplaretisoennsthrieplsatbioetnwsheiepns qbueatwnteiteinesq;ugaranptihties;
egqraupahtioenqsuaotniocnosoordnincoatoerdaixneastewaitxheslawbeitlhs alanbdelsscalneds.
sAclasloesN. A-Qls.oA.M2 AFS.912.N-Q.1.2

MP 1, MP 2, MP 3, MP 4, MP 7

Objective To write and graph an equation of a direct variation

As your distance The diagram shows how long it takes to hear thunder after you see
from lightning lightning. What general rule can you use to model this situation? Explain.
increases, so does
the time it takes 10 s 15 s
you to hear the 2 mi 3 mi
thunder.

MATHEMATICAL

PRACTICES

The time it takes to hear thunder varies directly with the distance from lightning.

Essential Understanding  If the ratio of two variables is constant, then the
variables have a special relationship, known as a direct variation.

Lesson A direct variation is a relationship that can be represented by a function in the

Vocabulary form y = kx, where k ≠ 0. The constant of variation for a direct variation k is the
• direct variation =
• constant of coefficient of x. By dividing each side of y kx by x, you can see that the ratio of the
y
variation for a variables is constant: x = k.
direct variation
To determine whether an equation represents a direct variation, solve it for y. If you can
write the equation in the form y = kx, where k ≠ 0, it represents a direct variation.

Problem 1 Identifying a Direct Variation

Do these equations Does the equation represent a direct variation? If so, find the constant of variation.
look like ones you’ve
seen before? A 7y = 2x B 3y + 4x = 8
Yes. They contain two
variables, so they’re y = 2 x d Solve each equation for y. S 3y = 8 - 4x
literal equations. To 7
determine whether 8 4
they’re direct variation The equation has the form y = kx, y = 3 - 3 x
equations, solve for y. so the equation is a direct variation.
Its constant of variation is 72. You cannot write the equation in the form

y = kx. It is not a direct variation.

Got It? 1. Does 4x + 5y = 0 represent a direct variation? If so, find the constant of variation.

Lesson 5-2  Direct Variation 301

To write an equation for a direct variation, first find the constant of variation k using an
ordered pair, other than (0, 0), that you know is a solution of the equation.

Problem 2 Writing a Direct Variation Equation

Suppose y varies directly with x, and y = 35 when x = 5. What direct variation
equation relates x and y? What is the value of y when x = 9?

Make sure you don’t stop y = kx Start with the function form of a direct variation.
at 7 = k. To write the 35 = k(5) Substitute 5 for x and 35 for y.
direct variation equation, 7=k Divide each side by 5 to solve for k.
Write an equation. Substitute 7 for k in y = kx.
you have to substitute y = 7x
7 for k in y = kx.
The equation y = 7x relates x and y. When x = 9, y = 7(9), or 63.

Got It? 2. Suppose y varies directly with x, and y = 10 when x = -2. What direct
variation equation relates x and y? What is the value of y when x = -15?

Problem 3 Graphing a Direct Variation STEM

Space Exploration  Weight on Mars y varies Weight on Mars
directly with weight on Earth x. The weights of the 50 lb
science instruments onboard the Phoenix Mars
Lander on Earth and Mars are shown.

A What is an equation that relates weight, in

pounds, on Earth x and weight on Mars y? Weight on Earth

y = kx Start with the function form of a direct variation. 130 lb

50 = k(130) Substitute 130 for x and 50 for y.

0.38 ≈ k Divide each side by 130 to solve for k.

y = 0.38x Write an equation. Substitute 0.38 for k in y = kx.

The equation y = 0.38x gives the weight y on Mars, in

Have you graphed pounds, of an object that weighs x pounds on Earth.

e quations like B What is the graph of the equation in part (A)?
y = 0.38x before? Make a table of values. Then draw the graph.
Yes. In Chapter 4, you

gb yramphaekdinlginaeatar bfulencotfions x y 60 y
values and plotting 40
points. 0 0.38(0) = 0 20 The points form a linear
pattern. Draw a line
50 0.38(50) = 19 through them.

100 0.38(100) = 38 x
O 50 100 150
150 0.38(150) = 57

302 Chapter 5  Linear Functions

Got It? 3. a. Weight on the moon y varies directly with weight on Earth x. A person
who weighs 100 lb on Earth weighs 16.6 lb on the moon. What is an

equation that relates weight on Earth x and weight on the moon y? What

is the graph of this equation?
b. Reasoning  What is the slope of the graph of y = 0.38x in Problem 3?

How is the slope related to the equation?

Concept Summary  Graphs of Direct Variations

The graph of a direct variation equation y = kx is y y
a line with the following properties. x x

• The line passes through (0, 0). kϽ0

• The slope of the line is k.

kϾ0

Y(xo,uy)cavnarryedwirrietcetalyd, ixyreisctthvearcioatniostnanetqoufavtiaorniaytio=nk. xItaiss y = k. When a set of data pairs
x same for each data pair.

the

Problem 4 Writing a Direct Variation From a Table

For the data in the table, does y vary directly with x? If it does, write an equation for
the direct variation.

A x y B x y
46 –2 3.2
8 12 1 2.4
10 15 4 1.6

Find y for each ordered pair. Find y for each ordered pair.
x x

6 = 1.5 12 = 1.5 15 = 1.5 3.2 = - 1.6 2.4 = 2.4 1.6 = 0.4
4 8 10 -2 1 4
y
The ratio y = 1.5 for each data pair. The ratio x is not the same for all data
x
pairs. So y does not vary directly
How can you check So y varies directly with x. The direct
your answer? with x.
variation equation is y = 1.5x.

Graph the ordered pairs

in the coordinate plane. xy
−3 2.25
I f you can connect them Got It? 4. For the data in the table at the right, does y vary
1 −0.75
with a line that passes directly with x? If it does, write an equation for the

through (0, 0), then y direct variation.
varies directly with x.

4 −3

Lesson 5-2  Direct Variation 303

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

1. Does the equation 6y = 18x represent a direct PRACTICES
variation? If it does, what is its constant of variation?
Vocabulary  Determine whether each statement is
always, sometimes, or never true.

2. Suppose y varies directly with x, and y = 30 when 5. The ordered pair (0, 0) is a solution of the direct
x = 3. What direct variation equation relates x and y? variation equation y = kx.

3. A recipe for 12 corn muffins calls for 1 cup of flour. 6. You can write a direct variation in the form
The number of muffins you can make varies directly y = k + x, where k ≠ 0.
with the amount of flour you use. You have 212 cups of
flour. How many muffins can you make? 7. The constant of variation for a direct variation
represented by y = kx is xy.
4. Does y vary directly with x? If it
does, what is an equation for the xy 8. Reasoning  Suppose q varies directly with p. Does
direct variation? –2 1 this imply that p varies directly with q? Explain.
2 –1
4 –2

Practice and Problem-Solving Exercises

A Practice Determine whether each equation represents a direct variation. If it does, find See Problem 1.
the constant of variation.

9. 2y = 5x + 1 10. 8x + 9y = 10 11. -12x = 6y
12. y + 8 = -x 13. -4 + 7x + 4 = 3y 14. 0.7x - 1.4y = 0

Suppose y varies directly with x. Write a direct variation equation that relates See Problem 2.

x and y. Then find the value of y when x = 12.

15. y = -10 when x = 2. 16. y = 712 when x = 3. 17. y = 5 when x = 2.
18. y = 125 when x = -5. 19. y = 10.4 when x = 4. 20. y = 913 when x = - 12.

Graph each direct variation equation. See Problem 3.

21. y = 2x 22. y = 1 x 23. y = -x 24. y = - 1 x
3 2

25. Travel Time  The distance d you bike varies directly with the amount of time t you
bike. Suppose you bike 13.2 mi in 1.25 h. What is an equation that relates d and t?
What is the graph of the equation?

26. Geometry  The perimeter p of a regular hexagon varies directly with the length / of
one side of the hexagon. What is an equation that relates p and /? What is the graph
of the equation?

304 Chapter 5  Linear Functions

For the data in each table, tell whether y varies directly with x. If it does, write an See Problem 4.
equation for the direct variation. Check your answer by plotting the points from
the table and sketching the line.

27. 28. y 29. y
xy x 5.4 x 1
3 12.6 6
−6 9 7 21.6 −2 11
1 −1.5 12 3
8 −12 8

B Apply Suppose y varies directly with x. Write a direct variation equation that relates
x and y. Then graph the equation.

30. y = 1 when x = 3. 31. y = - 5 when x = 14. 32. y = 6 when x = - 65. 33. y = 7.2 when x = 1.2.
2 5

34. Think About a Plan  The amount of blood in a person’s body varies directly with
body weight. A person who weighs 160 lb has about 4.6 qt of blood. About how
many quarts of blood are in the body of a 175-lb person?

• How can you find the constant of variation?
• Can you write an equation that relates quarts of blood to weight?
• How can you use the equation to determine the solution?

STEM 35. Electricity  Ohm’s Law V = I * R relates the voltage, current, and resistance of a
circuit. V is the voltage measured in volts. I is the current measured in amperes. R is
the resistance measured in ohms.

a. Find the voltage of a circuit with a current of 24 amperes and a resistance of 2 ohms.
b. Find the resistance of a circuit with a current of 24 amperes and a voltage of 18 volts.

Reasoning  Tell whether the two quantities vary directly. Explain
your reasoning.

36. the number of ounces of cereal and the number of Calories the cereal contains
37. the time it takes to travel a certain distance and the rate at which you travel
38. the perimeter of a square and the side length of the square
39. the amount of money you have left and the number of items you purchase

40. a. Graph the following direct variation equations in the same coordinate plane:
y = x, y = 2x, y = 3x, and y = 4x.

b. Look for a Pattern  Describe how the graphs change as the constant of

variation increases. 1
2
c. Predict how the graph of y = x would appear. xy
03
41. Error Analysis  Use the table at the right. A student says that y varies directly with 14
25
x because as x increases by 1, y also increases by 1. Explain the student’s error.

42. Writing  Suppose y varies directly with x. Explain how the value of y changes in

each situation. b. The value of x is halved.
a. The value of x is doubled.

Lesson 5-2  Direct Variation 305

STEM 43. Physics  The force you need to apply to a lever varies directly with the weight
you want to lift. Suppose you can lift a 50-lb weight by applying 20 lb of force to a
certain lever.

a. What is the ratio of force to weight for the lever?
b. Write an equation relating force and weight. What is the force you need to lift a

friend who weighs 130 lb?

C Challenge The ordered pairs in each exercise are for the same direct variation. Find each
missing value.

44. (3, 4) and (9, y) ( )45. (1, y) and 23, -9 46. ( -5, 3) and (x, -4.8)

47. Gas Mileage  A car gets 32 mi per gallon. The number of gallons g of gas used varies

directly with the number of miles m traveled.
a. Suppose the price of gas is $3.85 per gallon. Write a function giving the cost c

for g gallons of gas. Is this a direct variation? Explain your reasoning.
b. Write a direct variation equation relating the cost of gas to the miles traveled.
c. How much will it cost to buy gas for a 240-mi trip?

Standardized Test Prep

SAT/ACT 48. The price p you pay varies directly with the number of pencils you buy. Suppose Weekly Wages
you buy 3 pencils for $.51. How much is each pencil, in dollars?
Time Wages
49. A scooter can travel 72 mi per gallon of gasoline and holds 2.3 gal. The (h) ($)
function d(x) = 72x represents the distance d(x), in miles, that the scooter
can travel with x gallons of gasoline. How many miles can the scooter go with 12 99.00
a full tank of gas?
17 140.25
50. The table at the right shows the number of hours a clerk works per week and
the amount of money she earns before taxes. If she worked 34 h per week, how 21 173.25
much money would she earn, in dollars?
32 264.00
51. What is the greatest value in the range of y = x 2 - 3 for the domain
5-3, 0, 16?

Mixed Review

Find the slope of the line that passes through each pair of points. See Lesson 5-1.
55. (1, -2), ( -2, 3)
52. (2, 4), (0, 2) 53. (5, 8), ( -5, 8) 54. (0, 0), (3, 18)

Get Ready!  To prepare for Lesson 5-3, do Exercises 56–59.

Evaluate each expression. See Lesson 1-2.

56. 6a + 3 for a = 2 57. -2x - 5 for x = 3 58. 14 x + 2 for x = 16 59. 8 - 5n for n = 3

306 Chapter 5  Linear Functions

Concept Byte Investigating CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Use With Lesson 5-3 y = mx + b FM-BAFF.SB.931  2Id.Fen-BtiFfy.2t.h3e  Iedfefencttifoynthteheefgfreacpthonof
rtehpelagcrainpghfo (fx)rebpylaf c(xin)g+f k(x, )kbf (yx)f, (xf )(k+x)k, ,aknfd (x),
TECHNOLOGY f (xkx+), ka)nfdorf (sxp+eckif)icfovralsupeesciofifckv(abluoeths opfoskit(ibvoeth
panodsitniveegaatnivden) e.g. a. tive) . . .

MP 5

You can use a graphing calculator to explore the graph of an equation in the form MATHEMATICAL
y = mx + b. For this activity, choose a standard screen by pressing zoom 6.
PRACTICES

1. Graph these equations on the same screen. Then complete each statement.
1
y = x + 3 y = 2x + 3 y = 2 x + 3

a. The graph of ? is steepest.

b. The graph of ? is the least steep.

2. Match each equation with the best choice for its graph.

A. y = 14 x - 2 B. y = 4x - 2 C. y = x - 2
I. II. III.

3. Graph these equations on the same screen.

y = 2x + 3 y = -2x + 3

How does the sign of m affect the graph of the equation?

4. Reasoning  How does changing the value of m affect the graph of an equation in
the form y = mx + b?

5. Graph these equations on the same screen.

y = 2x + 3 y = 2x - 3 y = 2x + 2
Where does the graph of each equation cross the y-axis? (Hint: Use the ZOOM

feature to better see the points of intersection.)

6. Match each equation with the best choice for its graph.

A. y = 31 x - 3 B. y = 1 x + 1 C. y = 1 x
I. 3 3
II. III.

7. Reasoning  How does changing the value of b affect the graph of an equation in 307
the form y = mx + b?

Concept Byte  Investigating y = mx + b

5-3 Slope-Intercept Form CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

FM-IAFF.CS..79a1 2G.Fr-aIpFh.3l.i7naea  rGarnadphqulinaedarartaicndfuqnuctaidornastic
afunndctsihoonws aintderscheopwts,inmtearxciempats,,amndaxmiminai,maan.dAmlsionima.
A-lsSoSEM.AA.F1Sa.,9A1-2C.AED-S.ASE.2.1, .F1-aIF,.BM.4A,FFS-.L9E1.2A..A2-CED.1.2,
MMPAF1S,.M91P2.2F,-MIF.P2.34,, MMPAF4S.912.F-LE.1.2
MP 1, MP 2, MP 3, MP 4

Objectives To write linear equations using slope-intercept form
To graph linear equations in slope-intercept form

Each point of the Bamboo can grow very quickly. The graph models the Height (ft) Bamboo Growth
graph gives you growth of a bamboo plant. Find the point where the
information about line crosses the vertical axis. What does this point 40
the bamboo plant. tell you about the bamboo plant? Find the slope of
the line. What does the slope tell you about the 20
bamboo plant? How do you know?
0 0 10 20
Time (days)

MATHEMATICAL

PRACTICES

The function in the Solve It is a linear function, but it is not a direct variation. Direct

variations are only part of the family of linear functions.

A family of functions is a group of functions with common y

characteristics. A parent function is the simplest function with these 2 y‫؍‬x
characteristics. The linear parent function is y = x or f (x) = x. The y ‫ ؍‬؊21x ؉ 1 O x
graphs of three linear functions are shown at the right.
Lesson Ϫ2 2
A linear equation is an equation that models a linear function. In a
Vocabulary y ‫ ؍‬2x
• parent function
• linear parent linear equation, the variables cannot be raised to a power other than 1.
So y = 2x is a linear equation, but y = x2 and y = 2x are not. The
function
• linear equation graph of a linear equation contains all the ordered pairs that are
• y-intercept
• slope-intercept solutions of the equation. y-intercept: 0 y
Ϫ2 x
form Graphs of linear functions may cross the y-axis at any point. A
y-intercept of a graph is the y-coordinate of a point where the O2
graph crosses the y-axis.
Ϫ2

Essential Understanding  You can use the slope and y-intercept: Ϫ3
y-intercept of a line to write and graph an equation of the line.

Key Concept  Slope-Intercept Form of a Linear Equation

The slope-intercept form of a linear equation of a nonvertical line is y = mx + b.
c c
slope y@intercept

308 Chapter 5  Linear Functions

Problem 1 Identifying Slope and y-Intercept

What are the slope and y-intercept of the graph of y = 5x − 2?

Why isn’t the y = mx + b Use slope-intercept form.
y-intercept 2? slope y-intercept

In slope-intercept form, y = 5x + ( - 2) Think of y = 5x - 2 as y = 5x + (- 2).
The slope is 5; the y–intercept is -2.
the y-intercept b is added
to the term mx. Instead
of subtracting 2, you add

t he opposite, -2. Got It? 1. a. What are the slope and y–intercept of the graph of y = - 1 x + 2 ?
2 3
b. Reasoning  How do the graph of the line and the equation in part (a)

change if the y-intercept is moved down 3 units?

Problem 2 Writing an Equation in Slope-Intercept Form

When can you use What is an equation of the line with slope − 4 and y-intercept 7?
5

slope-intercept form? y = mx + b Use slope-intercept form.

You can write an equation y = - 4 x + 7 Substitute - 4 for m and 7 for b.
of a nonvertical line in 5 5
slope-intercept form if
you know its slope and An equation for the line is y = - 4 x + 7.
5

y-intercept. 3 -1?
Got It? 2. What 2
is an equation of the line with slope and y-intercept

Problem 3 Writing an Equation From a Graph

Multiple Choice  Which equation represents the line shown? 2y

y = -2x + 1 y = 1 x - 2 x
y = 2x + 1 2 2
W hat does the graph y = 2x - 2 Ϫ2 O
tell you about the
slope? Find the slope. Two points on the line are (0, -2) and (2, 2). Ϫ2
Since the line slants up
from left to right, the slope = 2 - ( - 2) = 4 = 2
slope of the line should 2 -0 2
be positive. The line is
also fairly steep, so the The y-intercept is -2. Write an equation in slope-intercept form.
slope of the line should
be greater than 1. y = mx + b

y = 2x + ( - 2) Substitute 2 for m and - 2 for b.

An equation for the line is y = 2x - 2. The correct answer is D.

Got It? 3. a. What do you expect the slope of the line to be from looking y x
at the graph? Explain.
1
b. What is an equation of the line shown at the right? Ϫ2 O 1
c. Reasoning  Does the equation of the line depend on the
Ϫ2
points you use to find the slope? Explain.

Lesson 5-3  Slope-Intercept Form 309

Problem 4 Writing an Equation From Two Points

What equation in slope-intercept form represents the line that passes through the
points (2, 1) and (5, −8)?

The line passes through An equation of the Use the two points to find the slope.
(2, 1) and (5, - 8). line Then use the slope and one point to
solve for the y-intercept.

Step 1 Use the two points to find the slope.

Can you use either slope = -8 - 1 = -9 = -3
point to find the 5-2 3
y-intercept?
Step 2 Use the slope and the coordinates of one of the points to find b.
Yes. You can substitute
y = mx + b Use slope-intercept form.
the slope and the
1 = - 3(2) + b Substitute - 3 for m, 2 for x, and 1 for y.
coordinates of any point
7 = b Solve for b.
on the line into the form
y = mx + b and solve Step 3 Substitute the slope and y-intercept into the slope-intercept form.
for b.

y = mx + b Use slope-intercept form.
y = - 3x + 7 Substitute - 3 for m and 7 for b.

An equation of the line is y = -3x + 7.

Got It? 4. What equation in slope-intercept form represents the line that passes
through the points (3, -2) and (1, -3)?

You can use the slope and y–intercept from an equation to graph a line.

What information can Problem 5 Graphing a Linear Equation
you use? What is the graph of y = 2x − 1?
The slope tells you the
ratio of vertical change y Step 2 The slope is 2, or 2 .
to horizontal change. Plot 21 1
the y-intercept. Then use 2x Move up 2 units and
the slope to plot another Ϫ2 O 2
point on the line. right 1 unit. Plot

another point.

Step 1 The y-intercept is Step 3 Draw a line through
Ϫ1. So plot a point the two points.
at (0, Ϫ1).

Got It? 5. What is the graph of each linear equation?
a. y = -3x + 4 b. y = 4x - 8

310 Chapter 5  Linear Functions

Slope-intercept form is useful for modeling real-life At 0 meters, the
situations where you are given a starting value (the pressure is 1 atm.
y-intercept) and a rate of change (the slope).
The pressure
Problem 6 Modeling a Function STEM increases by
0.1 atm/m.
How do you identify Physics  Water pressure can be measured in
the y-intercept? atmospheres (atm). Use the information in the Pressure Underwater
The y-intercept is the diagram to write an equation that models the 6
y-value when x = 0. pressure y at a depth of x meters. What graph 4
So the y-intercept is the models the pressure? 2
pressure at a depth of 00 20 40 60
0 m. This is the starting Step 1 Identify the slope and the y-intercept.
value, 1 atm. Depth (m)
The slope is the rate of change, 0.1 atm/m.
Pressure (atm)
The y-intercept is the starting value, 1 atm.

Step 2 Substitute the slope and y-intercept into the
slope-intercept form.

y = mx + b Use slope-intercept form.
y = 0.1x + 1 Substitute 0.1 for m and 1 for b.

Step 3 Graph the equation.

The y-intercept is 1. Plot the point (0, 1).
The slope is 0.1, which equals 110. Plot a

second point 1 unit above and 10 units to
the right of the y-intercept. Then draw a
line through the two points.

Got It? 6. A plumber charges a $65 fee for a repair
plus $35 per hour. Write an equation to
model the total cost y of a repair that takes
x hours. What graph models the total cost?

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES

1. What is an equation of the line with slope 6 and 4. Vocabulary  Is y = 5 a linear equation? Explain.
y-intercept -4?
5. Reasoning  Is it always, sometimes, or never true that
2. What equation in slope-intercept form represents an equation in slope-intercept form represents a
the line that passes through the points direct variation? Support your answer with examples.
( -3, 4) and (2, -1)?
6. Writing  Describe two different methods you can use
3. What is the graph of y = 5x + 2?
to graph the equation y = 2x + 4. Which method do
you prefer? Explain.

Lesson 5-3  Slope-Intercept Form 311

Practice and Problem-Solving Exercises

A Practice Find the slope and y-intercept of the graph of each equation. See Problem 1.
See Problem 2.
7. y = 3x + 1 8. y = -x + 4 9. y = 2x - 5
10. y = -3x + 2 11. y = 5x - 3
13. y = 4 14. y = -0.2x + 3 12. y = -6x

15. y = 1 x - 1
4 3

Write an equation in slope-intercept form of the line with the given
slope m and y-intercept b.

16. m = 1, b = -1 17. m = 3, b = 2 18. m = 21, b = - 1
2
8
19. m = 0.7, b = -2 20. m = -0.5, b = 1.5 21. m = -2, b = 5

Write an equation in slope-intercept form of each line. See Problem 3.

22. 4 y 23. y 24. y
2
x 2
2 Ϫ2 O 2 4 x
Ϫ2 Ϫ2 O 2
Ϫ2 O 2 x Ϫ2
Ϫ2

25. y 26. y 27. y
4
2 2
2 x x Ϫ2 O x
Ϫ2 O 2 Ϫ2 O 2 2
Ϫ2

Write an equation in slope-intercept form of the line that passes through the See Problem 4.

given points.

28. (0, 3) and (2, 5) 29. ( -2, 4) and (3, -1) 30. ( -3, 3) and (1, 2)

Graph each equation. See Problem 5.
See Problem 6.
31. y = x + 5 32. y = 3x + 4 33. y = -2x + 1

34. Retail Sales  Suppose you have a $5-off coupon at a fabric store. You buy
fabric that costs $7.50 per yard. Write an equation that models the total
amount of money y you pay if you buy x yards of fabric. What is the graph of the
equation?

35. Temperature  The temperature at sunrise is 65°F. Each hour during the day, the
temperature rises 5°F. Write an equation that models the temperature y, in degrees

Fahrenheit, after x hours during the day. What is the graph of the equation?

312 Chapter 5  Linear Functions

B Apply 36. Using the tables below, predict whether the two graphs will intersect. Plot the points
and sketch the lines. Do the two lines appear to intersect? Explain.

x y xy

Ϫ2 9 Ϫ2 Ϫ18
Ϫ1 7 Ϫ1 Ϫ14

05 0 Ϫ10

13 1 Ϫ6

21 2 Ϫ2

Find the slope and y-intercept of the graph of each equation.

37. y - 2 = -3x 38. y + 1 x = 0 39. y - 9x = 21 40. 2y - 6 = 3x
2 44. 2y + 4n = -6x

41. -2y = 6(5 - 3x) 42. y - d = cx 43. y = (2 - a)x + a

45. Think About a Plan  Polar bears are listed as a threatened species. In 2005, there

were about 25,000 polar bears in the world. If the number of polar bears declines by

1000 each year, in what year will polar bears become extinct?
• What equation models the number of polar bears?
• How can graphing the equation help you solve the problem?

46. Error Analysis  A student drew the graph at the right for the equation y
y = -2x + 1. What error did the student make? Draw the correct graph.

47. Computers  A computer repair service charges $50 for diagnosis and 2

$35 per hour for repairs. Let x be the number of hours it takes to repair x

a computer. Let y be the total cost of the repair. 2 0 2
a. Write an equation in slope-intercept form that relates x and y. 2
b. Graph the equation.
c. Reasoning  Explain why you should draw the line only in Quadrant I.

Use the slope and y-intercept to graph each equation.

48. y = 7 - 3x 49. 2y + 4x = 0 50. 3y + 6 = -2x

51. y + 2 = 5x - 4 52. 4x + 3y = 2x - 1 53. -2(3x + 4) + y = 0

Write a recursive formula and an explicit formula in slope-intercept form that
model each arithmetic sequence. How does the recursive formula relate to the
slope-intercept form?

54. 3, 5, 7, 9, . . . 55. -1, 3, 7, 11, . . . 56. 0.7, 0.3, -0.1, -0.5, . . .

57. Writing  Describe two ways you can determine whether an equation is linear.

58. Hobbies  Suppose you are doing a 5000-piece puzzle. You have already placed

175 pieces. Every minute you place 10 more pieces.

a. Write an equation in slope-intercept form to model the number of pieces placed.

Graph the equation.

b. After 50 more minutes, how many pieces will you have placed?

C Challenge Find the value of a such that the graph of the equation has the given slope m.

59. y = 2ax + 4, m = -1 60. y = - 1 ax - 5, m = 52 61. y = 3 ax + 3, m = 9
2 4 16

Lesson 5-3  Slope-Intercept Form 313

62. Sailing  A sailboat begins a voyage with 145 lb of food. The crew plans to eat a total

of 15 lb of food per day.
a. Write an equation in slope-intercept form relating the remaining food supply y

to the number of days x.
b. Graph your equation.
c. The crew plans to have 25 lb of food remaining when they end their voyage. How

many days does the crew expect their voyage to last?

Standardized Test Prep

SAT/ACT 63. Which equation represents the line that has slope 5 and passes through the
point (0, -2)?

y = x - 2 y = 5x - 2 y = -2x - 5 y = 5x

64. What is the slope of the line that passes through the points ( -5, 3) and (1, 7)?
5 2 2 3
- 3 - 3 3 2

65. Which number line shows the solution of 0 2x + 5 0 … 3?


Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5 Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5
Short
Response
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5 54321012345

66. Which equation represents the graph at the right? 5y

y = - 23x + 4 y = -23x + 4
3 2
y = -4x + 2 y = 4x - 3 2
Ϫ2 O
67. If a, b, and c are real numbers, a ≠ 0, and b 7 c, is the statement x
ab 7 ac always, sometimes, or never true? Explain. 246

Mixed Review

Suppose y varies directly with x. Write a direct variation equation that relates See Lesson 5-2.
x and y. Then find the value of y when x = 10.

68. y = 5 when x = 1. 69. y = 8 when x = 4. 70. y = 9 when x = 3.

Solve each equation. Justify each step. See Lesson 2-2.

71. 21 = -2t + 3 72. q - 3 = 6 73. 8x + 5 = 61
3

Get Ready!  To prepare for Lesson 5-4, do Exercises 74–77.

Simplify each expression. 76. - 94(x - 6) See Lesson 1-7.
74. -3(x - 5) 75. 5(x + 2) 77. 1.5(x + 12)

314 Chapter 5  Linear Functions

5-4 Point-Slope Form Common Core State Standards

MF-LAEC.CA.921  2C.oFn-sLtEru.1c.t2l inCeoanrs.t.ru. cfut nlicnteioanrs. . . f.ugnivcetinonas . . .
grivaepnh,aagdraepschr,ipatidoenscorfipatiroenlaotifoansrheliapt,iorntswhiop,inopr utwt-o
ionuptuptu-touptapirust. Apalsiros.AA-lCsoEDM.AA.C2C, F.9-I1F2.B.A.4-,CFE-DIF..1C.2.7,a,
FM-LAEC.CB.5912.F-IF.2.4, MACC.912.F-IF.3.7a,
MMAPC1C, .M91P23.F,-MLEP.24.5
MP 1, MP 3, MP 4

Objective To write and graph linear equations using point-slope form

Think about this Altitude (meters)The red line shows the altitude of a hot-air balloon during its linear
situation. The graph descent. What is an equation of the line in slope-intercept form? (Hint:
shows the altitude What is the altitude of the balloon when it starts its descent at x = 0 ?)
of the balloon with
respect to time. It y
doesn’t show the
path of the balloon. 680

MATHEMATICAL 640
(5, 640)
PRACTICES
600 (15, 620)

560 (35, 580)
(45, 560)

520

0 10 20 30 40 50 x
Time (seconds)

Lesson You have learned how to write an equation of a line by using its y-intercept. In this
lesson, you will learn how to write an equation without using the y-intercept.
Vocabulary
• point-slope form Essential Understanding  You can use the slope of a line and any point on the line to
write and graph an equation of the line. Any two equations for the same line are equivalent.

Key Concept  Point-Slope Form of a Linear Equation

Definition Graph
The point-slope form of an equation of a
nonvertical line with slope m and through y m ‫؍‬ rise When you use
point (x1, y1) is y - y1 = m(x - x1). run y Ϫ y1 ϭ m(x Ϫ x1),
(x1, y1) represents
Symbols (x1, y1)x a specific point and
y Ϫ y1 ϭ m(x Ϫ x1) O (x, y) represents
cc c any point.

y-coordinate slope x-coordinate

Lesson 5-4  Point-Slope Form 315

Here’s Why It Works  Given a point (x1, y1) on a line and the line’s slope m, you
can use the definition of slope to derive point-slope form.
y2 - y1
x2 - x1 = m Use the definition of slope.

# y - y1 = m Let (x, y) be any point on the line. Substitute (x, y) for (x2, y2).
x - x1 Multiply each side by (x - x1).
y - y1
x - x1 (x - x1) = m(x - x1)

y - y1 = m(x - x1) Simplify the left side of the equation.

Since you know a point Problem 1 Writing an Equation in Point-Slope Form
a nd the slope, use A line passes through (−3, 6) and has slope −5. What is an equation of the line?
point-slope form.
y - y1 = m(x - x1) Use point-slope form.

y1 ϭ 6 m ϭ Ϫ5 x1 ϭ Ϫ3

y - 6 = - 5[x - ( - 3)] Substitute (- 3, 6) for (x1, y1) and − 5 for m.
y - 6 = - 5(x + 3) Simplify inside grouping symbols.

Got It? 1. A line passes through (8, -4) and has slope 23. What is an equation in
point-slope form of the line?

Problem 2 Graphing Using Point-Slope Form

How does the What is the graph of the equation y − 1 = 2 (x − 2)?
equation help you 3
make a graph?
Use the point from the The equation is in point-slope form, y - y1 = m(x - x1). A point (x1, y1) on the line is
equation. Use the slope (2, 1), and the slope m is 32.
from the equation to find
another point. Graph Step 1 Graph a point Step 2 Use the slope, 2 . Go
using the two points. at (2, 1). 3
up 2 units and right

3 units. Draw a point.

y

43

22

O 24 x Step 3 Draw a line through
Ϫ2 the two points.

Got It? 2. What is the graph of the equation y + 7 = - 4 (x - 4)?
5

316 Chapter 5  Linear Functions

You can write the equation of a line given any two points on the line. First use the
two given points to find the slope. Then use the slope and one of the points to write
the equation.

Problem 3 Using Two Points to Write an Equation 4 y (1, 4)
What is an equation of the line at the right?
How does the graph
help you write an You need the slope m, m = y2 − y1 2 x
equation? so start with the slope x2 − x1 24
You can use two points formula. Ϫ4 Ϫ2 O
on the line to find the Use the given points to (؊2, ؊3) Ϫ2
slope. Then use find the slope.
point-slope form. Use point-slope form.

Use either given point for m = −3 − 4 = −7 = 7 Ϫ4
(x1, y1). For example, you −2 − 1 −3 3
can use (1, 4).
y − y1 = m(x − x1)

y − 4 = 7 (x − 1)
3

Got It? 3. a. In the last step of Problem 3, use the point ( -2, -3) instead of (1, 4) to
write an equation of the line.

b. Reasoning  Rewrite the equations in Problem 3 and part (a) in

slope-intercept form. Compare the two rewritten equations. What

can you conclude?

Problem 4 Using a Table to Write an Equation

How does the table Recreation  The table shows the altitude of a hot-air balloon Hot-Air Balloon Descent
help you write an during its linear descent. What equation in slope-intercept form Time, x Altitude, y
equation? gives the balloon’s altitude at any time? What do the slope and (s) (m)
The table gives four y-intercept represent? 10 640
points. You can use any 30 590
two of the points to m = 590 - 640 = - 2.5 U(3s0e, two points, such as (10, 640) and 70 490
find the slope. Then use 30 - 10 590), to find the slope. 90 440
point-slope form.
y - y1 = m(x - x1) Use point-slope form.

y - 640 = - 2.5(x - 10) sUlosepeth-e data point (10, 640) and the
2.5.

y = -2.5x + 665 Rewrite in slope-intercept form.

The slope -2.5 represents the rate of descent of the balloon in meters per second.
The y-intercept 665 represents the initial altitude of the balloon in meters.

Lesson 5-4  Point-Slope Form 317

Got It? 4. a. The table shows the number of gallons of water y Volume of Water in Tank
in a tank after x hours. The relationship is linear. Time, x Water, y
What is an equation in point-slope form that (h) (gal)
models the data? What does the slope represent? 2 3320
3 4570
b. Reasoning  Write the equation from part (a) 5 7070
in slope-intercept form. What does the 8 10,820
y-intercept represent?

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

1. What are the slope and one point on the graph of 5. Vocabulary  What features of the graph of the
4 equation y - y1 = m(x - x1) can you identify?
y - 12 = 9 (x + 7)?

2. What is an equation of the line that passes through 6. Reasoning  Is y - 4 = 3(x + 1) an equation of a line
the point (3, -8) and has slope -2? through ( -2, 1)? Explain.

3. What is the graph of the equation y - 4 = 3(x + 2)? 7. Reasoning  Can any equation in point-slope form
also be written in slope-intercept form? Give an
4. What is an equation of the line that passes through example to explain.
the points ( -1, -2) and (2, 4)?

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Write an equation in point-slope form of the line that passes through the given See Problem 1.
point and with the given slope m.
See Problem 2.
8. (3, -4); m = 6 9. (4, 2); m = - 5
10. ( -2, - 7); m = 54 3 See Problem 3.
4y
11. (4, 0); m = -1 2

Graph each equation. 13. y - 1 = -3(x + 2) x
12. y + 3 = 2(x - 1) O2
14. y + 5 = -(x + 2) 15. y - 2 = 4 (x - 3)
9

Write an equation in point-slope form for each line.

16. y 17. y 18.

23

x 1 x
O2 Ϫ2 O 2

Ϫ2

318 Chapter 5  Linear Functions

Write an equation in point-slope form of the line that passes through the given See Problem 4.

points. Then write the equation in slope-intercept form.

19. (1, 4), ( -1, 1) 20. (2, 4), ( -3, -6) 21. ( -6, 6), (3, 3)

Model the data in each table with a linear equation in slope-intercept form.
Then tell what the slope and y-intercept represent.

22. Time Volume of 23. Time Wages
Painting, x (days) Paint, y (gal) Worked, x (h) Earned, y ($)
2
3 56 1 8.50
5 44
20 3 25.50

6 51.00

B Apply Graph the line that passes through the given point and has the given slope m.
5 1
24. ( -3, -2); m = 2 25. (6, -1); m = - 3 26. ( -3, 1); m = 3

27. Think About a Plan  The relationship of degrees Fahrenheit (°F) and degrees Celsius

(°C) is linear. When the temperature is 50°F, it is 10°C. When the temperature is

77°F, it is 25°C. Write an equation giving the Celsius temperature C in terms of the

Fahrenheit temperature F. What is the Celsius temperature when it is 59°F?

• How can point-slope form help you write the equation?

• What are two points you can use to find the slope?

28. a. Geometry  Figure ABCD is a rectangle. Write equations in point-slope y B

form of the lines containing the sides of ABCD. 3 x
b. Reasoning  Make a conjecture about the slopes of parallel lines. A C
c. Use your conjecture to write an equation of the line that passes Ϫ4 O 1

through (0, -4) and is parallel to y - 9 = -7(x + 3). Ϫ2
D
STEM 29. Boiling Point  The relationship between altitude and the boiling point
of water is linear. At an altitude of 8000 ft, water boils at 197.6°F. At an
altitude of 4500 ft, water boils at 203.9°F. Write an equation giving the
boiling point b of water, in degrees Fahrenheit, in terms of the altitude a,

in feet. What is the boiling point of water at 2500 ft?

30. Using a graphing calculator, graph f (x) = 3x + 2.
a. If f (x) = 3x + 2 and g(x) = 4f (x), write the equation for g(x). Graph g(x) and

compare it to the graph of f (x).
b. If f (x) = 3x + 2 and h(x) = f (4x), write the equation for h(x). Graph h(x) and

compare it to the graph of f (x).

c. Compare how multiplying a function by a number and multiplying the x value of

a function by a number change the graphs of the functions.

31. Using a graphing calculator, graph f (x) = 2x - 5.
a. If f (x) = 2x - 5 and j(x) = f (x) + 3, write the equation for j(x). Graph j(x) and

compare it to the graph of f (x).
b. If f (x) = 2x - 5 and k(x) = f (x + 3), write the equation for k(x). Graph k(x) and

compare it to the graph of f (x).

c. Compare how adding a number to a function and adding a number to the x

value of a function change the graphs of the functions.

Lesson 5-4  Point-Slope Form 319

C Challenge 32. Forestry  A forester plants a tree and measures its circumference yearly over the
next four years. The table shows the forester’s measurements.

Tree Growth

Time (yr) 12 34

Circumference (in.) 2 4 6 8

a. Show that the data are linear, and write an equation that models the data.
b. Predict the circumference of the tree after 10 yr.
c. The circumference of the tree after 10 yr was actually 43 in. After four more

years, the circumference was 49 in. Based on this new information, does the

relationship between time and circumference continue to be linear? Explain.

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

MP 2, MP 4

In the Apply What You’ve Learned in Lesson 5-1, you showed that there is a

linear relationship between the number of marbles in the glass on page 293 and

the height of the water.

a. Choose two points from the table at the right, which shows Number of Height of
the height of the water in the glass when different numbers Marbles, x Water (cm), y
of marbles are dropped into it. Use the points to write an
equation in slope-intercept form that gives the water height 3 6.9
y as a function of the number of marbles x in the glass. 5 7.5
9 8.7
b. What does the y-intercept of the graph of your equation 14 10.2
represent? What does the slope represent? 17 11.1
23 12.9
c. Evaluate the function from part (a) when x = 40. Does the
water height given by the function make sense? Explain.
What does the function value tell you about what would
happen if you dropped 40 marbles in the glass?

320 Chapter 5  Linear Functions

5 Mid-Chapter Quiz MathX

OLMathXL® for School
R SCHO Go to PowerAlgebra.com
L®

FO

Do you know HOW? Write an equation in slope-intercept form of each line.

Each rate of change is constant. Find the rate of change 15. y 16. y
and explain what it means.
2 2
1. Studying for a Test 2. Distance a Car Travels x
Ϫ2 O Ϫ2 O 2
Study Time (s) Distance (m) Ϫ2 2x Ϫ2
Time (h) 3 75
Grade 6 150 Graph each equation.
5 85 9 225
6 87 12 17. y = 4x - 3
7 89 300 18. y + 3 = 12(x + 2)
8 91

Find the slope of the line that passes through each pair Write an equation in point-slope form for the line
of points.
through the given point and with the given slope m.
3. (7, 3), (5, 1) 4. ( -2, 1), (3, 6)
19. (2, - 2); m = - 1
5. (6, -4), (6, 6) 6. (2, 5), ( -8, 5) 2

20. (4, 0); m = 4

Tell whether each equation is a direct variation. If it is, Write an equation of the line that passes through each
find the constant of variation. pair of points.

7. y = 3x 21. (4, -2) and (8, -6)
8. 5x + 3 = 8y + 3 22. ( -1, -5) and (2, 10)
9. -3x - 35y = 14

Find the slope and y-intercept of the graph of each equation. Do you UNDERSTAND?
23. Writing  Describe two methods you can use to write
10. y = 1 x + 3
5 an equation of a line given its graph.

11. 3x + 4y = 12 24. Vocabulary  How can you find the y-intercept of the
graph of a linear equation?
12. 6y = -8x - 18
25. Reasoning  Can you graph a line if its slope is
13. Credit Cards  In 2000, people charged $1,243 billion undefined? Explain.
on the four most-used types of credit cards. In 2005,
people charged $1,838 billion on these same four 26. Business  A salesperson earns $18 per hour plus a
types of credit cards. What was the rate of change? $75 bonus for meeting her sales quota. Write and
graph an equation that represents her total earnings,
14. Bicycling  The distance a wheel moves forward including her bonus. What does the independent
varies directly with the number of rotations. Suppose variable represent? What does the dependent
the wheel moves 56 ft in 8 rotations. What distance variable represent?
does the wheel move in 20 rotations?

Chapter 5  Mid-Chapter Quiz 321

5-5 Standard Form MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

MA-ACFESD..9A1.2 .AC-rCeaEtDe.e1q.2ua  tCiorenastienetqwuoaotiromnsoirne tvwaroiaobrlems o. r.e.
gvararipahbleeqsu.a.t.iognraspohnecqouoartdiionnasteonaxceoso.rd. i.nAaltseoaxFe-sIF.B. .4,
AF-lIsFo.CM.7Aa,FFS-.9LE1.2A.F.2-I,FF.2-L.4E,.BM.5AFS.912.F-IF.3.7a,
MMPAF1S,.M91P2.2F,-MLEP.13.2, ,MMPA4FS.912.F-LE.2.5
MP 1, MP 2, MP 3, MP 4

Objectives To graph linear equations using intercepts
To write linear equations in standard form

A point on the graph An athlete wants to make a snack mix of peanuts Ounces of peanuts Snack Mix
tells you (ounces and cashews that will contain a certain amount of
of cashews, ounces protein. Cashews have 4 g of protein per ounce, 4
of peanuts). What and peanuts have 7 g of protein per ounce. 3
do you do with this How many grams of protein will the athlete’s mix 2
information to find contain? What do the points (7, 0) and (0, 4) 1
the grams of protein? represent? Explain. 00 1 2 3 4 5 6 7

Ounces of cashews

MATHEMATICAL In this lesson, you will learn to use intercepts to graph a line. Recall that a y-intercept
PRACTICES is the y-coordinate of a point where a graph crosses the y-axis. The x-intercept is the

x-coordinate of a point where a graph crosses the x-axis.

Lesson Essential Understanding  One form of a linear equation, called standard form,
allows you to find intercepts quickly. You can use the intercepts to draw the graph.
Vocabulary
• x-intercept Key Concept  Standard Form of a Linear Equation
• standard form of

a linear equation

The standard form of a linear equation is Ax + By = C, where A, B, and C are real
numbers, and A and B are not both zero.

Problem 1 Finding x- and y-Intercepts

Why do you What are the x- and y-intercepts of the graph of 3x + 4y = 24?
substitute 0 for y to
find the x-intercept? Step 1 To find the x-intercept, Step 2 To find the y-intercept,
The x-intercept is the substitute 0 for y. Solve for x. substitute 0 for x. Solve for y.
x-coordinate of a point
on the x-axis. Any point 3x + 4y = 24 3x + 4y = 24
on the x-axis has a 3x + 4(0) = 24 3(0) + 4y = 24
y-coordinate of 0. 3x = 24
x = 8 4y = 24
y=6
The x-intercept is 8.
The y-intercept is 6.

322 Chapter 5  Linear Functions

Got It? 1. What are the x- and y-intercepts of the graph of each equation?
a. 5x - 6y = 60 b. 3x + 8y = 12

Problem 2 Graphing a Line Using Intercepts
What is the graph of x − 2y = −2?

An equation of the line The coordinates of at Find and plot the x- and y-intercepts.
least two points on Draw a line through the points.
the line

Step 1 Find the intercepts. Step 2 Plot (Ϫ2, 0) and (0, 1). y
Draw a line through 2
x - 2y = -2 the points.
x - 2(0) = -2
What points do the x = -2 x
intercepts represent? x - 2y = -2
The x-intercept is - 2, 0 - 2y = -2 Ϫ2 O 2
so the graph crosses the -2y = -2 Ϫ2
x-axis at (- 2, 0). The y=1
y-intercept is 1, so the

graph crosses the y-axis

at (0, 1).

Got It? 2. What is the graph of 2x + 5y = 20?

If A = 0 in the standard form Ax + By = C, then you can write the equation in the
form y = b, where b is a constant. If B = 0, you can write the equation in the form
x = a, where a is a constant. The graph of y = b is a horizontal line, and the graph of
x = a is a vertical line.

Problem 3 Graphing Horizontal and Vertical Lines

What is the graph of each equation?

W hy write x = 3 in A x = 3 B y = 3
s tandard form? 1x + 0y = 3 d Write in standard form. S 0x + 1y = 3
Wi,n hsetannydoaurdwfroitrem,xy=ou3can y
y

see that, for any value 2 For all values 2
of y, x = 3. This form x of y, x ϭ 3.
of the equation makes x For all values
of x, y ϭ 3.
graphing the line easier. Ϫ2 O 2 Ϫ2 O 2

Ϫ2 Ϫ2

Got It? 3. What is the graph of each equation? d. y = 1
a. x = 4 b. x = -1 c. y = 0

Lesson 5-5  Standard Form 323

Given an equation in slope-intercept form or point-slope form, you can rewrite the
equation in standard form using only integers.

Problem 4 Transforming to Standard Form

What is y = − 3 x + 5 written in standard form using integers?
7

How can you get y = - 3 x + 5
started? 7
You need to clear the ( ) 3
fraction. So, multiply 7y = 7 - 7 x + 5 Multiply each side by 7.

each side of the equation 7y = - 3x + 35 Distributive Property
by the denominator of
3x + 7y = 35 Add 3x to each side.

the fraction. - = - 1 +
3
Got It? 4. Write y 2 (x 6) in standard form using integers.

Problem 5 Using Standard Form as a Model

Online Shopping  A media download store sells songs for $1 each and movies
for $12 each. You have $60 to spend. Write and graph an equation that describes
the items you can purchase. What are three combinations of numbers of
songs and movies you can purchase?

Relate cost of  times  number  plus  cost of  times  number  equals  $60
a song of songs a movie of

Define Let x = the number of songs purchased. movies

Let y # #= the number of movies purchased.
x + 12
Write 1 y = 60

Is there another way An equation for this situation is x + 12y = 60. 0 songs
to find solutions? Find the intercepts. 5 movies Downloads
You can guess and check
ob ynesuvbasritaitbulteinagndvasloulevsinfogr x + 12y = 60 x + 12y = 60 Number of Movies 5 24 songs
f or the other. Then check x + 12(0) = 60 0 + 12y = 60 4 3 movies
is feynosuerinsotlhuetioconnmteaxkteosf 3 60 songs
the problem. Graphing is x = 60 y=5 2 0 movies
the quickest way to see 1
all the solutions. Use the intercepts to draw the graph. Only points in the 00 12 24 36 48 60
first quadrant make sense.
Number of Songs
The intercepts give you two combinations of songs and
movies. Use the graph to identify a third combination.
Each of the red points is a possible solution.

Check for Reasonableness  You cannot buy a fraction of a song or movie. The graph is a
line, but only points with integer coordinates are solutions.

324 Chapter 5  Linear Functions

Got It? 5. a. In Problem 5, suppose the store charged $15 for each movie. What
equation describes the numbers of songs and movies you can purchase

for $60?
b. Reasoning  What domain and range are reasonable for the equation in

part (a)? Explain.

Concept Summary  Linear Equations

You can describe any line using one or more of these forms of a linear equation. Any two
equations for the same line are equivalent.

Graph Forms

y-intercept Slope-Intercept Form
(0, 6)
y y = m- 32xx++b6
y =
Slope
m ϭ Ϫ23 Point x-intercept Point-Slope Form
(3, 4) (9, 0)

2 x y - y41==-m23((xx--x31))
y -

Ϫ2 O 2 4 6 8 Standard Form
Ax + By = C
2x + 3y = 18

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

1. What are the x- and y-intercepts of the graph of PRACTICES
3x - 4y = 9?
6. Vocabulary  Tell whether each linear equation

is in slope-intercept form, point-slope form, or

2. What is the graph of 5x + 4y = 20? standard form.
a. y + 5 = -(x - 2)
3. Is the graph of y = -0.5 a horizontal line, a vertical b. y = -2x + 5
line, or neither? c. y - 10 = -2(x - 1)
d. 2x + 4y = 12
4. What is y = 1 x + 3 written in standard form
2
using integers? 7. Reasoning  Which form
would you use to write an 4y
5. A store sells gift cards in preset amounts. You can equation of the line at the x
purchase gift cards for $10 or $25. You have spent right: slope-intercept form, 1
$285 on gift cards. Write an equation in standard point-slope form, or standard O 24
form to represent this situation. What are three form? Explain.
combinations of gift cards you could have purchased?

Lesson 5-5  Standard Form 325

Practice and Problem-Solving Exercises

A Practice Find the x- and y-intercepts of the graph of each equation. See Problem 1.
10. -3x + 3y = 7
8. x + y = 9 9. x - 2y = 2 13. -5x + 3y = -7.5

11. 3x - 5y = -20 12. 7x - y = 21

Draw a line with the given intercepts. See Problem 2.

14. x-intercept: 3 15. x-intercept: -1 16. x-intercept: 4
y-intercept: 5 y-intercept: -4 y-intercept: -3

Graph each equation using x- and y-intercepts.

17. x + y = 4 18. x + y = -3 19. x - y = -8
22. 6x - 2y = 18
20. -2x + y = 8 21. -4x + y = -12

For each equation, tell whether its graph is a horizontal or a vertical line. See Problem 3.
26. x = -1.8
23. y = -4 24. x = 3 25. y = 74
30. x = 7
Graph each equation. 28. x = -3 29. y = -2
27. y = 6

Write each equation in standard form using integers. See Problem 4.

31. y = 2x + 5 32. y + 3 = 4(x - 1) 33. y - 4 = -2(x - 3)
34. y = 41 x - 2
35. y = - 2 x - 1 36. y + 2 = 2 (x + 4)
3 3

37. Video Games  In a video game, you earn 5 points for each jewel you find. You See Problem 5.
earn 2 points for each star you find. Write and graph an equation that represents
the numbers of jewels and stars you must find to earn 250 points. What are three
combinations of jewels and stars you can find that will earn you 250 points?

38. Clothing  A store sells T-shirts for $12 each and sweatshirts for $15 each. You

plan to spend $120 on T-shirts and sweatshirts. Write and graph an equation that

represents this situation. What are three combinations of T-shirts and sweatshirts

you can buy for $120?

B Apply 39. Writing  The three forms of linear equations you have studied are slope-intercept
form, point-slope form, and standard form. Explain when each form is most useful.

40. Think About a Plan  You are preparing a fruit salad. You want the total
carbohydrates from pineapple and watermelon to equal 24 g. Pineapple has 3 g of
carbohydrates per ounce and watermelon has 2 g of carbohydrates per ounce. What
is a graph that shows all possible combinations of ounces of pineapple and ounces
of watermelon?

• Can you write an equation to model the situation?
• What domain and range are reasonable for the graph?

326 Chapter 5  Linear Functions

41. Compare and Contrast  Graph 3x + y = 6, 3x - y = 6, and -3x + y = 6. How are
the graphs similar? How are they different?

42. Reasoning  What are the slope and y-intercept of the graph of Ax + By = C?

43. Error Analysis  A student says the equation y = 4x + 1 can be written in standard
form as 4x - y = 1. Describe and correct the student’s error.

44. Reasoning  The coefficients of x and y in the standard form of a linear equation
cannot both be zero. Explain why.

Graphing Calculator  Use a graphing calculator to graph each equation. Make a
sketch of the graph. Include the x- and y-intercepts.

45. 2x - 8y = -16 46. -3x - 4y = 0 47. x + 3.5y = 7

48. -x + 2y = -8 49. 3x + 3y = -15 50. 4x - 6y = 9

51. Compare and Contrast  The graph below represents one function, and the table
represents a different function. How are the functions similar? How are they
different?

y

2 x x Ϫ4 Ϫ2 0 2 4
24 y 5 43 2 1
Ϫ2 O
Ϫ2

Find the x- and y-intercepts of the line that passes through the given points.

52. ( -6, 4), (3, -5) 53. ( -5, -5), (4, -2) 54. ( -7, 6), ( -4, 11)

55. ( -2, 8), (4, 2) 56. (3, -8), ( -4, 13) 57. (5, 0.4), ( -1, -2)

58. Sports  The scoreboard for a football game is shown at the right. All of
the points the home team scored came from field goals worth 3 points
and touchdowns with successful extra-point attempts worth 7 points.
Write and graph a linear equation that represents this situation. List
every possible combination of field goals and touchdowns the team
could have scored.

C Challenge 59. Geometry  Graph x + 4y = 8, 4x - y = -1, x + 4y = -12, and 4x - y = 20
in the same coordinate plane. What figure do the four lines appear to form?

Write an equation of each line in standard form.
60. The line contains the point ( -4, -7) and has the same slope as the graph of

y + 3 = 5(x + 4).
61. The line has the same slope as 4x - y = 5 and the same y-intercept as the graph of

3y - 13x = 6.

Lesson 5-5  Standard Form 327

62. a. Graph 2x + 3y = 6, 2x + 3y = 12, and 2x + 3y = 18 in the same
coordinate plane.

b. How are the lines from part (a) related?
c. As C increases, what happens to the graph of 2x + 3y = C?

63. a. Fundraising  Suppose your school is having a talent show to raise money for
new band supplies. You think that 200 students and 150 adults will attend. It will
cost $200 to put on the talent show. What is an equation that describes the ticket
prices you can set for students and adults to raise $1000?

b. Open-Ended  Graph your equation. What are three possible prices you could set
for student and adult tickets?

Standardized Test Prep

SAT/ACT 64. What is y = - 3 x + 2 written in standard form using integers?
4
43 x + y = 2
3x + 4y = 2 3x + 4y = 8 -3x - 4y = 8
x - 2y = 4
65. Which of the following is an equation of a horizontal line?

3x + 6y = 0 2x + 7 = 0 -3y = 29

Short 66. Which equation models a line with the same y-intercept but half the slope of the
line y = 6 - 8x?
Response
y = -4x + 3 y = 6 - 4x y = 3 - 8x y = -16x + 6

67. What is the solution of   7 x - 19 = - 13 + 2x?
2

-9 -4 4 9

68. The drama club plans to attend a professional production. Between 10 and 15
students will go. Each ticket costs $25 plus a $2 surcharge. There is a one-time
handling fee of $3 for the entire order. What is a linear function that models this
situation? What domain and range are reasonable for the function?

Mixed Review

Write an equation in point-slope form of the line that passes through the given See Lesson 5-4.

points. Then write the equation in slope-intercept form.

69. (5, -1), ( -3, 4) 70. (0, -2), (3, 2) 71. ( -2, -1), (1, 2)

Solve each compound inequality. Graph your solution. See Lesson 3-6.
74. 3x + 1 7 10 or 5x + 3 … -2
72. -6 6 3t … 9 73. -9.5 6 3 - y … 1.3

Get Ready!  To prepare for Lesson 5-6, do Exercises 75–77.

Find the slope of the line that passes through each pair of points. See Lesson 5-1.

75. (0, -4), (2, 0) 76. (5, 5), (3, -1) 77. ( -4, 2), (5, 2)

328 Chapter 5  Linear Functions

Concept Byte Inverse of a CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Linear Function
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MP 3

Let f  be a function. If there is another function that pairs b with a whenever f pairs
a with b, then functions are called inverse functions.

You can find the inverse function algebraically.

Example

Find the inverse of the function f (x) = −2x + 4.

Step 1 Replace f(x) with y. Then switch x for y and y for x.

y = -2x + 4 Write f(x) as y.
x = -2y + 4 Switch x and y.

Step 2 Solve for y.

x - 4 = -2y Subtract 4 from each side.
--22y Divide by - 2.
x-4 = Simplify.
-2
- 12x + 2 = y

Step 3 Write in function notation, using f -1 to represent the inverse of the
function f.

f -1(x) = - 21x + 2

Exercises

Find the inverse of each function.

1. f (x) = 4x - 2 2. f (x) = 3x + 2 3. f (x) = -2x - 3
4. f (x) = 21x + 1
7. f (x) = 2 - 7x 5. f (x) = 4x + 8 6. f (x) = -4x
8. f (x) = 14x - 4 9. f (x) = - 32x + 1
10. f (x) = 2x + 8 11. f (x) = -4x + 8 12. f (x) = -1 - 2x
15. f (x) = 34x - 1
13. f (x) = -7 + x 14. f (x) = 4x - 5 18. f (x) = - 61x + 1
17. f (x) = - 2 - 32x
16. f (x) = x - 1

19. Reasoning  Why does interchanging the x and y match the definition of an inverse function?

Concept Byte  Inverse of a Linear Function 329

5-6 Parallel and MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Perpendicular Lines
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pgreoobmleemtrsic. problems.

MP 1, MP 2, MP 3, MP 4, MP 7

Objectives To determine whether lines are parallel, perpendicular, or neither
To write equations of parallel lines and perpendicular lines

Copy the graph shown at the right. Can you 4y
draw a line that will not intersect either of
Shift your the lines in the graph? If so, draw the line. Ϫ2 O 4x
perspective to If not, why not? Ϫ2
include the area
outside the grid Can you draw a line that will intersect
as well. one of the lines in such a way that the
intersection forms four congruent angles?
MATHEMATICAL If so, draw the line. If not, why not?

PRACTICES

Lesson Two distinct lines in a coordinate plane either intersect or are parallel. Parallel lines are
lines in the same plane that never intersect.
Vocabulary
• parallel lines Essential Understanding  You can determine the relationship between two lines
• perpendicular by comparing their slopes and y-intercepts.

lines
• opposite

reciprocals

Key Concept  Slopes of Parallel Lines

Words Graph
Nonvertical lines are parallel if they have the same
slope and different y-intercepts. Vertical lines are 2y
parallel if they have different x-intercepts.

Ϫ2 O 4x

Example 1 sxlo+p1e,a21n,danyd=d21ifxfe-re2ntayre-inlinteerscepts.
2
The graphs of y = Ϫ4

that have the same

The lines are parallel.

You can use the fact that the slopes of parallel lines are the same to write the equation
of a line parallel to a given line.

330 Chapter 5  Linear Functions

Problem 1 Writing an Equation of a Parallel Line

Why start with A line passes through (12, 5) and is parallel to the graph of y = 2 x − 1. What
point-slope form? 3
You know a point on the equation represents the line in slope-intercept form?
line. You can use what
you know about parallel Step 1 Identify the slope of the given line. The slope of the graph of y = 2 x - 1 is 32.
lines to find the slope. 3
So, point-slope form is The parallel line has the same slope.
convenient to use.
Step 2 Write an equation in slope-intercept form of the line through (12, 5) with
slope 32.

y - y1 = m(x - x1) Start with point-slope form.

y - 5 = 2 (x - 12) Substitute (12, 5) for (x1, y1) and 2 for m.
3 3

y - 5 = 2 x - 2 (12) Distributive Property
3 3

y - 5 = 2 x - 8 Simplify.
3

y = 2 x - 3 Add 5 to each side.
3

The graph of y = 2 x - 3 passes through (12, 5) and is parallel to the graph of
y= 2 3
3 x - 1.

Got It? 1. A line passes through (-3, -1) and is parallel to the graph of y = 2x + 3.
What equation represents the line in slope-intercept form?

You can also use slope to determine whether two lines are perpendicular.
Perpendicular lines are lines that intersect to form right angles.

Key Concept  Slopes of Perpendicular Lines

Words Graph

Two nonvertical lines are perpendicular if the product y
of their slopes is -1. A vertical line and a horizontal
line are also perpendicular. 1 x
4
Ϫ2 O 2
Ϫ2
Example
1 21.
The graph of y = 2 x - 1 has a slope of

The graph of y = -2x + 1 has a slope of -2.
1 - = - 1,
Since 2 ( 2) the lines are perpendicular.

Two numbers whose product is - 1 are opposite reciprocals. So, the slopes of

perpendicular lines are opposite reciprocals. To find the opposite reciprocal
#oSifn-ce43,-fo43r tfhinedotphpeorseicteiprreoccipalr,o-ca43l.oTfh-en43.write 43.
example, first its opposite,
4 4
3 = - 1, 3 is

Lesson 5-6  Parallel and Perpendicular Lines 331

Problem 2 Classifying Lines

Why write each Are the graphs of 4y = − 5x + 12 and y = 4 x − 8 parallel, perpendicular, or
equation in slope- 5
intercept form? neither? Explain.
You can easily identify
the slope of an equation Step 1 Find the slope of each line by writing its equation in slope-intercept form, if
in slope-intercept necessary. Only the first equation needs to be rewritten.
form. Just look at the
coefficient of x. 4y = - 5x + 12 Write the first equation.

4y = - 5x + 12 Divide each side by 4.
4 4

y = - 5 x + 3 Simplify.
4

The slope of the graph of y = - 5 x + 3 is - 45.
4

The slope of the graph of y = 4 x - 8 is 54.
5

Step 2 The slopes are not the same, so the lines cannot be parallel. Multiply the

#slopes to see if they are opposite reciprocals.-54=-1
4 5

The slopes are opposite reciprocals, so the lines are perpendicular.

Got It? 2. Are the graphs of the equations parallel, perpendicular, or neither? Explain.
a. y = 34 x + 7 and 4x - 3y = 9 b. 6y 1
= -x + 6 and y = - 6 x + 6

Problem 3 Writing an Equation of a Perpendicular Line

Multiple Choice  Which equation represents the line that passes through (2, 4) and
1
How do you know is perpendicular to the graph of y = 3 x − 1?
you have found the
opposite reciprocal? y = 1 x + 10 y = 3x + 10 y = -3x - 2 y = -3x + 10
Multiply the two 3
numbers together as a
check. If the product is Step 1 Identify the slope of the graph of the given equation. The slope is 31.
- 1, the numbers are
opposite reciprocals: Step 2 Find the opposite reciprocal of the slope from Step 1. The opposite reciprocal
13( - 3) = - 1. 1
of 3 is - 3. So, the perpendicular line has a slope of - 3.

Step 3 Use point-slope form to write an equation of the perpendicular line.

y - y1 = m(x - x1) Write point-slope form.
y - 4 = -3(x - 2) Substitute (2, 4) for (x1, y1) and − 3 for m.
y - 4 = -3x + 6 Distributive Property
y = -3x + 10 Add 4 to each side.

The equation is y = -3x + 10. The correct answer is D.

Got It? 3. A line passes through (1, 8) and is perpendicular to the graph of
y = 2x + 1. What equation represents the line in slope-intercept form?

332 Chapter 5  Linear Functions

Problem 4 Solving a Real-World Problem STEM y CEILING VIEW
12
Have you seen a Architecture  An architect uses software to design
problem like this the ceiling of a room. The architect needs to enter an 10
before? equation that represents a new beam. The new beam
Yes. You wrote will be perpendicular to the existing beam, which is 8
the equation of a represented by the red line. The new beam will pass
perpendicular line in through the corner represented by the blue point.
Problem 3. Follow the What is an equation that represents the new beam?
same steps here after
you calculate the slope of Step 1 Use the slope formula to find the slope of the 6
the line from the graph. red line that represents the existing beam.
4 Existing
m = 4 - 6 Points (3, 6) and (6, 4) 2 beam
6 - 3 are on the red line.
x
= - 2 Simplify. O 2 4 68 10 12
3

The slope of the line that represents the
existing beam is - 32.

Step 2 Find the opposite reciprocal of the slope 2 32.
3
from Step 1. The opposite reciprocal of - is

Step 3 Use point-slope form to write an equation. The

slope of the line that represents the new beam is
ro32e.r,pItirnewssiellolnpptsaes-thisnetthenrreocwuepgbhtefa(o1mr2m,i,1s0yy)=.-A321nx0e-=qu823.a(txio-n
that
12)

Got It? 4. What equation could the architect enter to represent a second beam whose
graph will pass through the corner at (0, 10) and be parallel to the existing
beam? Give your answer in slope-intercept form.

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

1. Which equations below have graphs that are 4. Vocabulary  Tell whether the two numbers in each

parallel to one another? Which have graphs that are pair are opposite reciprocals.

perpendicular to one another? a. - 2, 1 b. 1 , 4 c. 5, -5
2 4
y = - 1 x y = 6x y = 6x - 2
6 5. Open-Ended  Write equations of two parallel lines.

2. What is an equation of the line that passes through 6. Compare and Contrast  How is determining if two
lines are parallel similar to determining if they are
(3, -1) and is parallel to y = -4x + 1? Give your perpendicular? How are the processes different?
answer in slope-intercept form.

3. What is an equation of the line that passes through

(2, -3) and is perpendicular to y = x - 5? Give your
answer in slope-intercept form.

Lesson 5-6  Parallel and Perpendicular Lines 333

Practice and Problem-Solving Exercises

A Practice Write an equation in slope-intercept form of the line that passes through the See Problem 1.
given point and is parallel to the graph of the given equation.

7. (1, 3); y = 3x + 2 8. (2, -2); y = -x - 2 9. (1, -3); y + 2 = 4(x - 1)
10. (2, -1); y = - 32 x + 6 12. (4, 2); x = -3
11. (0, 0); y = 2 x + 1
3

Determine whether the graphs of the given equations are parallel, See Problem 2.

perpendicular, or neither. Explain. 15. y = -2x + 3
2x + y = 7
13. y = x + 11 14. y = 3 x - 1
y = -x + 2 4 18. y = 4x - 2
3 -x + 4y = 0
y = 4 x + 29

16. y - 4 = 3(x + 2) 17. y = -7
2x + 6y = 10 x = 2

Write an equation in slope-intercept form of the line that passes through the See Problem 3.

given point and is perpendicular to the graph of the given equation.

19. (0, 0); y = -3x + 2 20. ( - 2, 3); y = 1 x - 1 21. (1, -2); y = 5x + 4
2

22. ( -3, 2); x - 2y = 7 23. (5, 0); y + 1 = 2(x - 3) 24. (1, -6); x - 2y = 4

25. Urban Planning  A path for a new city park will y See Problem 4.
connect the park entrance to Main Street. The
path should be perpendicular to Main Street. 6 Park
What is an equation that represents the path? entrance
Main Street
4

26. Bike Path  A bike path is being planned for 2

the park in Exercise 25. The bike path will be

parallel to Main Street and will pass through the O2 x

park entrance. What is an equation of the line

that represents the bike path?

B Apply 27. Identify each pair of parallel lines. Then identify each pair of perpendicular lines. 1
2
line a: y = 3x + 3 line b: x = -1 line c: y - 5 = (x - 2)

line d: y = 3 line e: y + 4 = -2(x + 6) line f : 9x - 3y = 5

Determine whether each statement is always, sometimes, or never true. Explain.

28. A horizontal line is parallel to the x-axis.

29. Two lines with positive slopes are parallel.

30. Two lines with the same slope and different y-intercepts are perpendicular.

31. Reasoning  For an arithmetic sequence, the first term is A(1) = 3. Each successive

term adds 2 to the previous term. Another arithmetic sequence has the

rule B(n) = 5 + (n - 1)d, where n is the term number and d is the

common difference. If the graphs of the two sequences are parallel,

what is the value of d? Explain. x Ϫ1 0 12
y Ϫ1 3 7 11
32. Reasoning  Will the graph of the line represented by the table
intersect the graph of y = 4x + 5? Explain.

334 Chapter 5  Linear Functions

33. Think About a Plan  A designer is creating a new logo, as shown at the right. y x
The designer wants to add a line to the logo that will be perpendicular to the blue 6
line and pass through the red point. What equation represents the new line? 4 2 4
2
• What is the slope of the blue line?
• What is the slope of the new line? O

34. Reasoning  For what value of k are the graphs of 12y = -3x + 8 and
6y = kx - 5 parallel? For what value of k are they perpendicular?

35. Agriculture  Two farmers use combines to harvest corn from their fields.

One farmer has 600 acres of corn, and the other has 1000 acres of corn. Each

farmer’s combine can harvest 100 acres per day. Write two equations for the

number of acres y of corn not harvested after x days. Are the graphs of the equations

parallel, perpendicular, or neither? How do you know?

C Challenge 36. Geometry  In a rectangle, opposite sides are parallel and adjacent sides are
perpendicular. Figure ABCD has vertices A( -3, 3), B( -1, -2), C(4, 0), and D(2, 5).
Show that ABCD is a rectangle.

37. Geometry  A right triangle has two sides that are perpendicular to each other.

Triangle PQR has vertices P(4, 3), Q(2, -1), and R(0, 1). Determine whether PQR is
a right triangle. Explain your reasoning.

Standardized Test Prep

SAT/ACT 38. Which equation represents the graph of a line parallel to the line at the right? 2y
O
y = 1 x + 5 y = -2x + 4 x
2 Ϫ2 2
Ϫ2
y = 2x - 6 y = - 1 x - 2
2
Short
Response 39. What is the solution of (5x - 1) + ( -2x + 7) = 9? 3 5
37 1

40. Sal’s Supermarket sells cases of twenty-four 12-oz bottles of water for $15.50.
Shopper’s World sells 12-packs of 12-oz bottles of water for $8.15. Which store has
the better price per bottle? Explain.

Mixed Review

Graph each equation using x- and y-intercepts. See Lesson 5-5.
43. x - 3y = -6
41. x + y = 8 42. 2x + y = -3

Get Ready!  To prepare for Lesson 5-7, do Exercises 44–47.

Write an equation in slope-intercept form of the line that passes through the See Lesson 5-3.
given points. 47. (13, 20), (6, 60)

44. (1, 1), (3, 7) 45. (2, 5), (12, 1) 46. (0.5, 2), (4.5, 3)

Lesson 5-6  Parallel and Perpendicular Lines 335

5-7 Scatter Plots CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
and Trend Lines
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sausgcgaettsetsr palolint ethaartasusogcgieasttiosna. lAinlesaor Sa-sIsDo.cBia.6tiaon, .S-ID.C.7,
AS-lIsDo.CM.8A,FSS-.I9D1.C2.S9-ID.2.6a, MAFS.912.S-ID.3.7,
MMAPF1S,.M91P2.2S,-MIDP.33.8, ,MMPA4FS.912.S-ID.3.9
MP 1, MP 2, MP 3, MP 4

Objectives To write an equation of a trend line and of a line of best fit
To use a trend line and a line of best fit to make predictions

Music Sales

The table shows the number of digital Albums
albums downloaded per year and the number Downloaded CDs Sold
of CDs sold by manufacturers per year. Year (millions) (millions)
What relationship exists between the two
Would you expect sets of data? Predict the number of CDs 2004 4.6 767
the number of sold and the number of albums downloaded
albums downloaded in 2010. Explain your reasoning. 2005 13.6 705.4
to have an effect
on CD sales? 2006 27.6 619.7

MATHEMATICAL 2007 42.5 511.1

PRACTICES SOURCE: Recording Industry Association of America

Lesson In the Solve It, the number of albums downloaded per year and the number of CDs sold
per year are related.
Vocabulary
• scatter plot Essential Understanding  You can determine whether two sets of numerical
• positive data are related by graphing them as ordered pairs. If the two sets of data are related,
you may be able to use a line to estimate or predict values.
correlation
• negative A scatter plot is a graph that relates two different sets of data by displaying them
as ordered pairs. Most scatter plots are in the first quadrant of the coordinate plane
correlation because the data are usually positive numbers.
• no correlation
• trend line You can use scatter plots to find trends in data. The scatter plots below show the three
• interpolation types of relationships that two sets of data may have.
• extrapolation
• line of best fit yyy
• correlation

coefficient
• causation

0x 0x 0x

When y tends to increase as x When y tends to decrease as x When x and y are not related,
increases, the two sets of data increases, the two sets of data the two sets of data have
have a positive correlation. have a negative correlation. no correlation.

336 Chapter 5  Linear Functions

Problem 1 Making a Scatter Plot and Describing Its Correlation

Temperature  The table shows the altitude of an airplane and the temperature
outside the plane.

Altitude (m) 0 Plane Altitude and Outside Temperature
Temperature 59.0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
(°F) 59.2 61.3 55.5 41.6 29.8 29.9 18.1 26.2 12.4 0.6

T he highest altitude is A Make a scatter plot of the data. Plane Altitude and
Outside Temperature
5 000 m. So a reasonable Treat the data as ordered pairs. For the altitude of 60
scale on the altitude axis 40
is 0 to 5500 with every 1500 m and the temperature of 55.5°F, plot (1500, 55.5). 20Temperature (؇F)
00 1000
1 000 m labeled. You can B What type of relationship does the scatter plot show? 2000
Altitude (m)3000
ula axsbeise.slitmhielatremrepaesorantiunrgeto tThheeatletmitupdeeraotfutrheeopultasnideeinthcerepalsaense. tends to decrease as 4000
So the data have a 58 4 5000
1.3 2.2 1.1
negative correlation.

Got It? 1. a. Make a scatter plot of the data in the table
below. What type of relationship does the
scatter plot show?


Gasoline Purchases

Dollars Spent 10 11 9 10 13

Gallons Bought 2.5 2.8 2.3 2.6 3.3

b. Reasoning  Consider the population of a city and the number of  letters
in the name of the city. Would you expect a positive correlation, a
negative correlation, or no correlation between the two sets of data?
Explain your reasoning.

When two sets of data have a positive or negative y y
correlation, you can use a trend line to show the

correlation more clearly. A trend line is a line on

a scatter plot, drawn near the points, that shows

a correlation. 0 x0 x
You can use a trend line to estimate a value between

two known data values or to predict a value outside the range of known data values.

Interpolation is estimating a value between two known values. Extrapolation is

predicting a value outside the range of known values.

Lesson 5-7  Scatter Plots and Trend Lines 337

Problem 2 Writing an Equation of a Trend Line STEM

How do you draw an Biology  Make a scatter plot of the data at the right. What is the Weight of a Panda
accurate trend line? approximate weight of a 7-month-old panda? Age Weight
An accurate trend line
should fit the data Step 1 Make a scatter plot and draw a trend line. Estimate the (months) (lb)
closely. There should be coordinates of two points on the line. 1 2.5
about the same number 2 7.6
of points above the line Weight of a Panda
as below it.
Weight (lb) 3 12.5
How can you check 48 4 17.1
the reasonableness 36 6 24.3
of your answer?
Since x = 7 is visible 24 8 37.9
on the graph, find its 10 49.2
corresponding y-value. 12 12 54.9
When x = 7, y ≈ 32.7.
So the estimate is 00 2 4 6 8 10 12
reasonable. Age (months)

Two points on the trend line are (4, 17.1) and (8, 37.9).

Step 2 Write an equation of the trend line.

m = y2 - y1 = 37.9 - 17.1 = 20.8 = 5.2 Find the slope of the trend line.
x2 - x1 8 - 4 4 Use point-slope form.
Substitute 5.2 for m and (4, 17.1) for (x1, y1).
y - y1 = m(x - x1) Distributive Property
Add 17.1 to each side.
y - 17.1 = 5.2(x - 4)

y - 17.1 = 5.2x - 20.8

y = 5.2x - 3.7

Step 3 Estimate the weight of a 7-month-old panda.

y = 5.2(7) - 3.7 Substitute 7 for x.

y = 32.7 Simplify.

The weight of a 7-month-old panda is about 32.7 lb.

Got It? 2. a. Make a scatter plot of the data below. Draw a trend line and write its
equation. What is the approximate body length of a 7-month-old panda?

Body Length of a Panda
Age (months) 1 2 3 4 5 6 8 9

Body Length (in.) 8.0 11.75 15.5 16.7 20.1 22.2 26.5 29.0

b. Reasoning  Do you think you can use your model to extrapolate the body
length of a 3-year-old panda? Explain.

338 Chapter 5  Linear Functions

The trend line that shows the relationship between two sets of data most accurately is
called the line of best fit. A graphing calculator computes the equation of the line of
best fit using a method called linear regression.

The graphing calculator also gives you the r ‫ ؍‬؊1 r‫؍‬0 r‫؍‬1

correlation coefficient r, a number from strong negative no strong positive
-1 to 1, that tells you how closely the correlation correlation correlation
equation models the data.

The nearer r is to 1 or -1, the more closely the data cluster around the line of best fit.
If r is near 1, the data lie close to a line of best fit with positive slope. If r is near -1,
the data lie close to a line of best fit with negative slope.

Problem 3 Finding the Line of Best Fit

College Tuition  Use a graphing calculator to find the equation Average Tuition and Fees
of the line of best fit for the data at the right. What is the at Public 4-Year Colleges
correlation coefficient to three decimal places? Predict the
cost of attending in the 2012–2013 academic year. Academic Cost
Year ($)

Step 1 Press stat . From the EDIT menu, choose Edit. Enter 2000–2001 3508
Step 2 the years into L1. Let x = 2000 represent academic year
2000–2001, x = 2001 represent 2001–2002, and so on. 2001–2002 3766
Enter the costs into L2.
2002–2003 4098
Press stat . Choose LinReg(ax + b) from the CALC
menu. Press enter to find the equation of the line of 2003–2004 4645
best fit and the correlation coefficient. The calculator
uses the form y = ax + b for the equation. 2004–2005 5126

2005–2006 5492

2006–2007 5836

slope SOURCE: The College Board

LinReg y-intercept
y ϭ ax ϩ b
a ϭ 409.4285714 correlation
b ϭ Ϫ815446.7143 coefficient
r2 ϭ .9919891076
r ϭ .9959864997

What does the value Round to the nearest hundredth. The equation of the line of best fit is
of the correlation y = 409.43x - 815,446.71. The correlation coefficient is about 0.996.
coefficient mean?
The correlation Step 3 Predict the cost of attending in the 2012–2013 academic year.
coefficient of 0.996
is close to 1. So there y = 409.43x - 815,446.71 Use the equation of the line of best fit.
is a strong positive
correlation between the y = 409.43(2012) - 815,446.71 Substitute 2012 for x.
academic year and the
cost of attending college. y ≈ 8326 Simplify. Round to the nearest whole number.

The cost of attending a four-year public college in the 2012–2013 academic year is
predicted to be about $8326.

Lesson 5-7  Scatter Plots and Trend Lines 339

Got It? 3. a. Predict the cost of attending in the 2016–2017 academic year.
b. Reasoning  What does the slope of the line of best fit in Problem 3 tell

you about the rate of change in the cost?

Causation is when a change in one quantity causes a change in a second quantity.
A correlation between quantities does not always imply causation.

Problem 4 Identifying Whether Relationships Are Causal

In the following situations, is there likely to be a correlation? If so, does the
correlation reflect a causal relationship? Explain.

A the number of loaves of bread baked and the amount of flour used

There is a positive correlation and also a causal relationship. As the number of loaves of

Causal relationships bread baked increases, the amount of flour used increases.

a lways have a B the number of mailboxes and the number of firefighters in a city
ct wororedlaattiaonse. tHsotwhaetvehra,ve a There is likely to be a positive correlation because both the number of mailboxes
and the number of firefighters tend to increase as the population of a city increases.
correlation may not have
However, installing more mailboxes will not cause the number of firefighters to
a causal relationship.

increase, so there is no causal relationship.

Got It? 4. In the following situations, is there likely to be a correlation? If so, does the
correlation reflect a causal relationship? Explain.

a. the cost of a family’s vacation and the size of their house
b. the time spent exercising and the number of Calories burned

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Use the table.
4. Vocabulary  Given a set of data pairs, how would you

Average Maximum Daily Temperature decide whether to use interpolation or extrapolation
in January for Northern Latitudes
to find a certain value?

Latitude (° N) 35 33 30 25 43 40 39 5. Compare and Contrast  How are a trend line and the
44 line of best fit for a set of data pairs similar? How are
Temperature (°F) 46 52 67 76 32 37 they different?

SOURCE: U.S. Department of Commerce 6. Error Analysis  Refer to the table below. A student
says that the data have a negative correlation
1. Make a scatter plot of the data. What type of because as x decreases, y also decreases. What is the
relationship does the scatter plot show? student’s error?

2. Draw a trend line and write its equation. x 10 7 5 41 0

3. Predict the average maximum daily temperature in y 1 0 −2 −4 −7 −9
January at a latitude of 50° N.

340 Chapter 5  Linear Functions