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4-6
Graphing and Writing Linear
Functions
Going Deeper
Essential question: How can you represent relationships using linear functions?
Math Background
Standards for
Mathematical Content
A linear function of the form f (x) = mx + b can
be written if the slope and the y-intercept are
known. The slope can be determined as long as
the coordinates of two points are known or can be
obtained from a graph. Calculate the slope from the
coordinates of two points as follows:
(x2) - f (x1)
rise f____________
m = ____
run = x2 - x1 .
You can then find the value of b using m and
one of the known points. If the known point is
(x1, f (x1)), then b = f (x1) - mx1, or
f (x2) - f (x1)
b = f (x1) - ____________
x2 - x1 (x1). The function
f (x) = mx + b can then be rewritten as
f (x) = mx + b =
f____________
(x2) - f (x1)
f____________
(x2) - f (x1)
x2 - x1 (x) + f (x1) - x2 - x1 (x1).
A-CED.1.2 Create equations in two … variables
to represent relationships between quantities;
graph equations on coordinate axes with labels and
scales.*
A-REI.4.11 Explain why the x-coordinates of the
points where the graphs of the equations
y = f (x) and y = g(x) intersect are the solutions of
the equation f (x) = g(x); … Include cases where
f (x) and/or g(x) are linear … functions.*
F-IF.2.4 For a function that models a relationship
between two quantities, interpret key features of
graphs and tables in terms of the quantities, and
sketch graphs showing key features given a verbal
description of the relationship.*
F-IF.2.6 Calculate and interpret the average rate of
change of a function (presented symbolically or as a
table) over a specified interval. Estimate the rate of
change from a graph.*
F-IF.3.7 Graph functions expressed symbolically
and show key features of the graph ...*
F-IF.3.7a Graph linear ... functions and show
intercepts ...*
F-BF.1.1 Write a function that describes a
relationship between two quantities.*
F-LE.1.2 Construct linear … functions, … given a
graph, a description of a relationship, or two inputoutput pairs (include reading these from
a table).*
IN
= . TR OD UCE
Ask students to graph the function f(x) = 3x + 4 on
graph paper and identify the slope and the y-intercept.
Have them identify several points of the graph and
verify that each pair of points has the same slope.
1
EXAMPLE
Questioning Strategies
• Does the line in the graph for part A have positive
or negative slope? negative
Prerequisites
• Does the line in the graph for part A rise or fall as
the run moves from left to right? fall
Continuous linear functions
Slope values of m and b in f(x) = mx + b
• Why does the graph in part B have a y-intercept
of 1? The pitcher starts out containing 1 cup
of juice.
continued
Chapter 4
215
Lesson 6
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TE ACH
Name
Class
Notes
4-6
Date
Graphing and Writing Linear Functions
Going Deeper
Essential question: How can you represent relationships using linear functions?
Graphing Lines You can graph the linear function f (x) = mx + b using only the
slope m and the y-intercept b. First, locate the point (0, b) on the y-axis. Next, use
the rise and run of the slope to locate another point on the line. Draw the line
through the two points.
When using m to locate a second point, bear in mind that m is a ratio, so many values of
1 , then you could use a rise of __
1 and a run
rise and run are possible. For instance, if m = __
2
2
of 1, a rise of 1 and run of 2, a rise of -2 and a run of -4, and so on. Your choice of rise
and run often depends on the scales used on the coordinate plane’s axes.
F-IF.3.7a
1
EXAMPLE
Graphing a Line Using the Slope and y -Intercept
Graph each function.
2x + 4
f (x) = -__
3
A
4 . Plot the point that
r The y-intercept is
corresponds to the y-intercept.
r The slope is
3
run is
2
-_
3
6
.
4
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r Use the slope to move from the first point to a second
2
point. Begin by moving down
units, because the
3
rise is negative. Then move right
units, because
the run is positive. Plot a second point.
2
x
-4
real
2
4
6
8
-4
numbers.
real
The range is the set of
0
-2
-2
r Draw the line through the two points.
r The domain is the set of
B
f(x)
8
. If you use -2 as the rise, then the
numbers.
Amount of liquid (cups)
A pitcher with a maximum capacity of 4 cups contains 1 cup of apple juice
concentrate. A faucet is turned on filling the pitcher at a rate of 0.25 cup
per second. The amount of liquid in the pitcher (in cups) is a function A(t) of
the time t (in seconds) that the water is running.
r The y-intercept is the initial amount in the pitcher at
1
time 0, or
cup. Plot the point that corresponds
to the y-intercept.
r The slope is the rate of change: 0.25 cup per second,
4
1
or 1 cup in
seconds. So, the rise is
and
4 .
the run is
8
A(t)
6
4
2
t
0
r Use the rise and run to move from the first point to a
1
second point on the line by moving up
unit and
4
right
units. Plot a second point.
4
8
12 16
Time (seconds)
continued
215
Chapter 4
Lesson 6
r Connect the points and extend the line to the maximum
4
value of the function, where A(t) =
cups.
1
The range is the set of numbers
≤t≤
12 .
≤ A(t) ≤
4
0
r The domain is the set of numbers
.
1a. How could you use the slope and y-intercept to graph the function f (x) = 3?
Write the rule as f(x) = 0x + 3. Start at the point (0, 3). The slope m = 0, so the rise
is 0 when the run is 1. Plot (1, 3) and draw a horizontal line through (0, 3) and (1, 3).
1b. What are the units of the rise in Part B? What are the units of the run?
The units of the rise are cups. The units of the run are seconds.
1c. How long does it take to fill the pitcher? Explain.
12 seconds; the input associated with a maximum output of 4 cups is 12 seconds.
1 t + 1? Use units to justify your answer.
1d. Why is the function rule A(t) = __
4
Sample answer: If the function rule is written as f(x) = mx + b, then m is the
rate of change and b is the y-intercept, or the initial value where x = 0. Since the
pitcher starts with 1 cup of water at time 0, b = 1 cup. The rate of change is
cups
1
0.25 cup per second = _
cup per second, and _____ · seconds + cups = cups.
4
2
second
F-LE.1.2
EXAMPLE
Writing a Linear Function
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REFLECT
Write the linear function f using the given information.
A
The graph of the function has a slope of 3 and a y-intercept of -1.
A linear function has the form f (x) = mx + b where m is the slope and
b is the y-intercept. Substitute
So, the function is f (x) =
B
3
for m and
-1
A different function has the values shown in the table.
First calculate the slope using any two ordered pairs
from the table. For instance, let ( x1, f ( x1 ) ) = (-1, 5) and
( x2, f ( x2 ) ) = (3, -3).
f ( x2 ) - f ( x1 )
m = ____________
x -x
2
1
5
-3
-
3
( -1 )
= _____________
Chapter 4
Chapter 4
-
for b.
3x - 1 .
x
f(x)
-1
5
3
-3
7
-11
Write the slope formula.
Substitute values.
216
Lesson 6
216
Lesson 6
1 EXAMPLE
2
continued
Questioning Strategies
• Why is the graph in part B a segment, not a line?
The pitcher starts with 1 cup in it at t = 0 and
reaches full capacity at t = 12. The graph is
Questioning Strategies
• For part B, do you get the same slope no matter
which two ordered pairs you choose? yes
• For part B, what would be the linear function if
you used known points (-1, 5) and (7, -11)?
f(x) = -2x + 3
limited by the 4-cup capacity of the pitcher.
Technology
Students may benefit from using a graphing
calculator to see the graph in part A. Ask them
2 X + 4 and then verify that the
to graph Y1 = - __
3
graph matches the one in the example. Also ask
them to verify that (0, 4) and (3, 2) are points of
the graph either by using the Trace feature or by
looking at points in the Table function.
Avoid Common Errors
When calculating the slope using ordered pairs
with negative coordinates, students will sometimes
forget the negative sign when subtracting values.
For example, for (-1, 5) and (3, -3), x2 - x1 should
be 3 - (-1), not 3 - 1. Remind students that they
must retain any negative signs in an ordered pair
when using the slope formula.
()
Avoid Common Errors
Students may have difficulty graphing a function
that has a fractional rate of change as shown in
this Example. Remind them that the fraction can
be looked at as rise over run, so the rise and run
can be used to move from one point to a second
point. That is, if the rate of change is expressed as a
fraction, use the numerator to move the appropriate
number of units up or down (the rise) and use the
denominator to move the appropriate number of
units right (the run) to plot a second point given the
first point.
Teaching Strategy
Encourage students to use the third point in the
table to check that they have the correct function.
EXTRA EXAMPLE
Write the linear function f using the given
information.
A. The graph of the function has a slope of -5 and
a y-intercept of 1. f (x) = -5x + 1
B. The function has values shown in the table.
2
x
f(x)
-1
-2
1
6
3
14
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EXTRA EXAMPLE
Graph each function.
A. f (x) = -__1 x - 3 The graph is a line with
2
1 . The domain is all
y-intercept -3 and slope -__
real numbers.
EXAMPLE
f(x) = 4x + 2
B. An oil storage tank with a volume of 6000 gallons
starts off full and empties at a rate of 60 gallons
per minute. The graph is a line with y-intercept
6000 and slope -60. The maximum value is
6000, and the minimum value is 0. The domain
is 0 ≤ x ≤ 100.
Chapter 4
217
Lesson 6
Notes
-8
= ______
Simplify numerator and denominator.
= -2
Simplify fraction.
4
Then find the value of b using the fact that m =
f (x) = -2 x + b
5 = -2
-2
and f (-1) = 5.
Write the function with the known value of m.
( -1 ) + b
Substitute -1 for x and 5 for f(x).
5 = 2 +b
Simplify the right side of the equation.
3 =b
Solve for b.
So, the function is f (x) = -2x + 3 .
REFLECT
2a. In Part B, use the ordered pair (7, -11) to check your answer.
f(7) = -2(7) + 3 = -14 + 3 = -11
F-LE.1.2
3
EXAMPLE
Writing a Linear Function from a Graph
The graph shows the increase in pressure (measured in
pounds per square inch) as a scuba diver descends from
a depth of 10 feet to a depth of 30 feet.
What is the pressure on the diver at the water’s surface?
Pressure (Ib/in.2)
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Pressure is the result of the weight of the column of
water above the diver as well as the weight of the
column of Earth’s atmosphere above the water. Pressure
is a linear function of depth.
Scuba Dividing
32
P(d)
(30, 28.0)
24
(10, 19.1)
16
8
d
A
0
Interpret the question.
10
20
30
Depth (feet)
Let d represent depth and P represent pressure. At the
water’s surface,
0
d=
. For this value of d, what meaning does P(d) have in terms
of the line that contains the line segment shown on the graph?
It is the line’s P-intercept.
Find the value of m in P(d) = md + b. Use the fact that P(10) = 19.1 and P(30) = 28.0.
B
P( d2 ) - P( d1 )
m = _____________
d2 - d1
Write the slope formula.
28.0 - 19.1
= ________________
30 - 10
Substitute values.
continued
217
Chapter 4
8.9
= ______
Simplify numerator and denominator.
= 0.445
Write in decimal form.
20
Find the value of b in P(d) = md + b. Use the value of m from Part B as well as
the fact that P(10) = 19.1.
P(d) =
0.445 d + b
19.1 = 0.445
19.1 =
14.7 ≈ b
4.45
(
Write the function with the known value of m.
)
10 + b
+b
Substitute 10 for d and 19.1 for P(d).
Simplify the right side of the equation.
Solve for b. Round to the nearest tenth.
So, the pressure at the water’s surface is P(0) = b ≈
14.7 lb/in.2
REFLECT
3a. Interpret the value of m in the context of the problem.
m is the rate of change in pressure with respect to depth.
3b. Write the function P(d) = md + b using the calculated values of m and b. Use the
function to find the pressure on the diver at a depth of 20 feet.
P(d) = 0.445d + 14.7; P(20) = 23.6 lb/in.2
4
A-REI.4.11
EXAMPLE
Writing and Solving a System of Equations
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© Houghton Mifflin Harcourt Publishing Company
C
Lesson 6
Mr. Jackson takes a commuter bus from his suburban home to his job in the city.
He normally gets on the bus in the town where he lives, but today he is running
a little late. He gets to the bus stop 2 minutes after the bus has left. He wants to
catch up with the bus by the time it gets to the next stop in a neighboring town
5 miles away.
The speed limit on the road connecting the two stops is 40 miles per hour, but
Mr. Jackson knows that the bus travels the road at 30 miles per hour. He decides
to drive at 40 miles per hour to the next stop. Does he successfully catch the
bus there?
A
Identify the independent and dependent variables, how they are measured, and how
you will represent them.
time
The independent variable is
, measured in minutes. Let t represent the
time since Mr. Jackson began driving to the next bus stop.
The dependent variable is distance , measured in miles. Let d represent the
distance traveled. Since you need to track the distances traveled by both Mr. Jackson
and the bus, use subscripts: dJ will represent the distance traveled by Mr. Jackson, and
dB will represent the distance traveled by the bus.
Chapter 4
Chapter 4
218
Lesson 6
218
Lesson 6
3
EXAMPLE
Highlighting
the Standards
Questioning Strategies
• Why can you extend the graph so that it intersects
the y-axis? The depths of the diver from 0 to
3 EXAMPLE is a real-world problem that
includes opportunities for mathematical
modeling, reasoning, and computation. It is
a good opportunity to address Mathematical
Practice Standard 4 (Model with mathematics).
Draw students’ attention to the way they
interpret the values of m and b. The value of
m gives the rate of change in pressure with
respect to depth: The pressure on the diver
increases 0.445 lb/in.2 for every additional
foot of depth. The value of b gives the
pressure on the diver at the water’s surface,
which is just the pressure due to the weight of
the column of Earth’s atmosphere above the
water: The pressure at the water’s surface is
14.7 lb/in.2
10 feet are reasonable to include in the domain of
the function.
• What would be the pressure at a depth of 40 feet?
Explain how you found your answer. 32.5 lb/in.2;
P(40) = 0.445(40) + 14.7 = 32.5
• Why is the graph a first-quadrant graph only? Only
nonnegative domain and range values make sense
in this situation.
Technology
Have students substitute the function in Reflect
Question 3b into a graphing calculator and then
use the Table feature to find additional points on
the graph.
Weight (kilograms)
EXTRA EXAMPLE
The graph shows the weight of a female elephant as
she grows from 1 to 3 years. What was the weight of
the elephant at birth?
4
w
900
Questioning Strategies
• What do the coordinates of the intersection point
represent in the context of this situation? The
(3, 800.0)
700
500
300
distance at which Mr. Jackson catches up with the
bus (4 miles) and the time it took Mr. Jackson
to reach that distance (6 minutes).
(1, 333.3)
100
• Why are conversions necessary in part B?
a
1
2
Age (years)
The speeds given are in miles per hour, but the
elapsed time is in minutes. Both measures need
to use the same unit of time to solve and graph
the functions.
3
100 kg
• Why is each graph only in the first quadrant? Only
nonnegative domain and range values make sense
in this situation.
EXTRA EXAMPLE
You are participating in a 2-mile run for charity. You
are at the back of the group of runners and leave the
starting line 0.5 minute after the first runners have
left. You would like to catch up to the first runners
by the time you reach the halfway point of the race
at 1 mile. The first runners run at a rate of 7.2 miles
per hour, and you run at a rate of 7.5 miles per hour.
Do you successfully catch up to the lead runners by
the time you reach the halfway point of the event?
No; you catch up 12 minutes after the race starts,
when both you and the lead runners have run
1.5 miles.
Chapter 4
219
Lesson 6
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0
EXAMPLE
Notes
B
Write a distance-traveled function for Mr. Jackson and for the bus.
Each function has the form d(t) = rt + d0 where r is the rate of travel and d0 is any
initial distance. Although you know the rates of travel, they are given in miles per
hour, which is incompatible with the unit of time (minutes). So, you need to convert
miles per hour to miles per minute. In the conversions below, express the miles as
simplified fractions.
40 miles · ___________
1 hour = _
2 mile per minute
Mr. Jackson: ________
60 minutes
hour
3
30 miles · ___________
1 hour = _
1 mile per minute
Bus: ________
60 minutes
hour
2
At the moment Mr. Jackson begins driving to the next bus stop, the bus
has traveled for 2 minutes. If you use Mr. Jackson’s position as the
starting point, then the initial distance for Mr. Jackson is 0 miles, and the
_1
initial distance for the bus is 2 · 2 = 1 mile .
So, the distance-traveled functions are:
2 t+
Mr. Jackson: dj(t) = _
0
3
6
d(t)
(6, 4)
4
2
t
0
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2
Determine the value of t for which dJ(t) = dB(t). You can do this by graphing
the two functions and seeing where the graphs intersect. Carefully draw the
graphs on the coordinate plane below, and label the intersection point.
Distance (miles)
C
1 t+
Bus: dB(t) = _
1
2
4
6
8
Time (minutes)
6
The t-coordinate of the point of intersection is
, so
Mr. Jackson catches up with the bus in 6 minutes .
D
Check the result against the conditions of the problem, and then answer
the problem’s question.
5
The problem states that the next bus stop is
miles away, and the
4
graph shows that Mr. Jackson catches up with the bus in
So, does Mr. Jackson successfully catch the bus?
miles.
Yes
REFLECT
4a. Explain how you can use algebra rather than a graph to find the time when
Mr. Jackson catches up with the bus. Then show that you get the same result.
1
2
1
Solve the equation: _
t+1=_
t: 1 = _
t, so t = 6.
2
3
6
219
Chapter 4
Lesson 6
4b. In terms of the context of the problem, explain why the t-coordinate of the
intersection point (and not some other point) determines how long it takes
Mr. Jackson to catch up with the bus.
The two graphs are distance-traveled graphs, and their intersection point is the
only point on the two graphs where the distances are equal. The t-coordinate of
PRACTICE
Graph each linear function.
1x + 2
2. f (x) = __
2
1. f (x) = 3x - 4
f(x)
f(x)
4
4
2
2
x
-4
-2
0
2
x
4
-4
-2
0
-2
-2
-4
-4
2
4
2
4
2
4
4x
4. f (x) = __
3
3. f (x) = -1
f(x)
f(x)
4
4
2
2
x
-4
-2
0
2
x
4
-4
-2
-2
-2
-4
-4
1x-3
5. f (x) = __
4
6. f (x) = -5x + 1
f(x)
f(x)
4
4
2
2
x
-4
Chapter 4
Chapter 4
-2
0
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
this point gives the time at which this occurs.
2
x
4
-4
-2
0
-2
-2
-4
-4
220
Lesson 6
220
Lesson 6
CLOSE
PR ACTICE
Essential Question
How can you represent relationships using linear
functions? You can represent a linear relationship
Where skills are
taught
graphically or symbolically. If you have the function
rule, you can draw the graph using the slope and
y-intercept. To write a rule from a graph, find the
y-intercept b and use two points on the graph to
find the value of m; then substitute those values
into f (x) = mx + b. From a verbal description,
identify the independent and dependent variables,
determine the coordinates of points that fit the
situation, and write the function f(x) = mx + b.
From a table of values, use two ordered pairs to
find the value of m and then use m and an ordered
pair to find b; then substitute those values into
f (x) = mx + b.
Where skills are
practiced
1 EXAMPLE
EXS. 1–6
2 EXAMPLE
EXS. 9–16
3 EXAMPLE
EXS. 17–19
4 EXAMPLE
EX. 20
Exercises 7–8: Students use the skills taught in the
lesson to graph a function and extend these skills to
make a prediction about the function.
Summarize
Have students complete a graphic organizer
showing the steps for writing a linear function from
two given points. A sample is shown below.
Given 2
points:
Solve for the
slope m.
Substitute m
and the
coordinates of
one point into
f(x) = mx + b.
Solve for the
y-intercept b.
Substitute m
and b into
f(x) = mx + b.
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Chapter 4
221
Lesson 6
Notes
Graph each linear function and answer the question. Explain your answer.
300
8. A bamboo plant is 10 centimeters tall at
noon and grows at a rate of 5 centimeters
every 2 hours. The height (in centimeters) is
a function h(t) of the time t it grows. When
will the plant be 20 centimeters tall?
C(t)
h(t)
30
Height (cm)
Cost (dollars)
7. A plumber charges $50 for a service call
plus $75 per hour. The total of these costs
(in dollars) is a function C(t) of the time
t (in hours) on the job. For how many hours
will the cost be $200? $300?
200
100
20
10
t
0
4
2
t
0
6
4
2
Time (hours)
6
Time (hours)
2 hours; the input value associated with
about 4:00 P.M.; the input value
an output of $200 is 2 hours; about
associated with an output of 20 cm is
3.3 hours; the input associated with an
about 4 hours, and 4 hours after noon
output of $300 is about 3.3 hours.
is 4:00 P.M.
Write the linear function f using the given information.
9. The graph of the function has a slope
of 4 and a y-intercept of 1.
10. The graph of the function has a slope
of 0 and a y-intercept of 6.
f(x) = 6
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f(x) = 4x + 1
12. The graph of the function has a slope
7 and a y-intercept of 0.
of __
4
11. The graph of the function has a slope
2 and a y-intercept of 5.
of -__
3
7
x
f(x) = _
2
f(x) = -_
x+5
4
3
13.
14.
x
f(x)
x
f(x)
−3
8
0
−3
0
5
2
0
3
2
4
3
3
x-3
f(x) = _
f(x) = -x + 5
15.
2
16.
x
f(x)
1
−1
x
5
f(x)
−2
2
5
10
−6
3
11
15
−10
4
f(x) = -_
x+2
f(x) = 6x - 7
5
221
Chapter 4
Lesson 6
Write the linear function f using the given information.
17.
18.
y
8
y
4
6
2
4
x
-4
0
-2
2
4
2
x
0
-2
-4
2
4
6
-2
1
f(x) = -_
x+3
f(x) = 2x - 1
2
19. The graph shows the amount of gas remaining in the gas tank
of Mrs. Liu’s car as she drives at a steady speed for 2 hours.
How long can she drive before her car runs out of gas?
Gas Remaining (gallons)
Fuel Consumption
a. Interpret the question by describing what aspect of the
graph would answer the question.
Want to know the t-intercept if the graph were
extended to the t-axis.
g(t)
16
12
8
4
b. Write a linear function whose graph includes the segment shown.
t
0
1
2
3
Time (hours)
g(t) = -2t + 10
c. Tell how to use the function to answer the question; then find the answer.
Set g(t) equal to 0 and solve for t; she can drive for 5 hours.
20. Jamal and Nathan exercise by running one circuit of a basically circular route that
is 5 miles long and takes them past each other’s home. The two boys run in the
same direction, and Jamal passes Nathan’s home 12 minutes into his run. Jamal
runs at a rate of 7.5 miles per hour while Nathan runs at a rate of 6 miles per hour.
If the two boys start running at the same time, when, if ever, will Jamal catch up
with Nathan before completing his run?
© Houghton Mifflin Harcourt Publishing Company
a. Identify the independent and dependent variables, how they are measured,
and how you will represent them.
Independent variable is time t measured in
10
minutes; dependent variable is distance d
measured in miles.
b. Write distance-run functions for Jamal and Nathan.
1
1
dj(t) = _
t; dN(t) = __
t + 1.5
8
10
Distance (miles)
© Houghton Mifflin Harcourt Publishing Company
-2
d(t)
8
(60, 7.5)
6
4
2
c. Graph the functions, find the intersection point, and
check the point against the conditions of the problem
to answer the question.
t
0
20
40
60
80
Time (minutes)
Jamal does not catch up with Nathan.
Chapter 4
Chapter 4
222
Lesson 6
222
Lesson 6
4 x-4
6. y = _
3
ADD I T I O N A L P R AC T I C E
AND PRO BL E M S O LV I N G
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
Answers
Additional Practice
1. y = 4x - 3
2. y = -2x
1 x+6
3. y = -_
3
2 10 + 5
4. 3 = _
5
5 x-5
7. y = _
2
( )
3=4+b
-1 = b
( )
2 x-1
y= _
5
5. y = -x + 3
b. slope: 3; number of hours per week;
y-int: 22; hours already worked
c. 70 hours
Problem Solving
1. y = 10x + 300
2. slope: 10, rate of the change of the cost:
$10 per student; y-int: 300, the initial fee
(the cost for 0 students)
3. $800
Chapter 4
223
4. C
5. J
6. A
7. H
Lesson 6
© Houghton Mifflin Harcourt Publishing Company
8. a. y = 3x + 22
Name
Class
Notes
4-6
Date
Additional Practice
© Houghton Mifflin Harcourt Publishing Company
223
Chapter 4
Lesson 6
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Problem Solving
Chapter 4
Chapter 4
224
Lesson 6
224
Lesson 6