168 Chapter 4 Functions
Comparing Linear and
Nonlinear Functions
4.4
STATE
STANDARDS
MA.8.A.1.6
S
How can you recognize when a pattern
in real life is linear or nonlinear?
Work with a partner. Copy and complete each table for the sequence of
similar rectangles. Graph the data in each table. Decide whether each pattern
is linear or nonlinear.
x
2x
a. Perimeters of Similar Rectangles b. Areas of Similar Rectangles
x 1 2 3 4 5
P
x 1 2 3 4 5
A
x
P
20
10
0
40
30
42 89631075
x
A
20
10
0
40
30
42 89631075
ACTIVITY: Finding Patterns for Similar Figures
1
MSFL8PE_0404.indd 168 10/20/09 4:24:16 PM
English
Spanish
Section 4.4 Comparing Linear and Nonlinear Functions 169
Work with a partner. The table shows the height h (in feet) of a falling object
at t seconds.
● Graph the data in the table.
● Decide whether the graph is linear or nonlinear.
● Compare the two falling objects. Which one has an increasing speed?
a. Falling parachute jumper b. Falling bowling ball
t 01234
h 300 285 270 255 240
t 01234
h 300 284 236 156 44
t
h
120
60
0
240
180
4289631075
Time (seconds)
Height (feet)
Parachute Jumper
Bowling Ball
t
h
120
60
0
240
180
4289631075
Time (seconds)
Height (feet)
ACTIVITY: Comparing Linear and Nonlinear Functions
2
2
Use what you learned about comparing linear and nonlinear
functions to complete Exercises 3 – 6 on page 172.
3. IN YOUR OWN WORDS How can you recognize when a pattern in real life
is linear or nonlinear? Describe two real-life patterns: one that is linear
and one that is nonlinear. Use patterns that are different from those
described in Activities 1 and 2.
English
Spanish
170 Chapter 4 Functions
Lesson
4.4
Lesson Tutorials
The graph of a linear function shows a constant rate of change. A
nonlinear function does not have a constant rate of change. So, its
graph is not a line.
EXAMPLE
Identifying Functions from Tables
1
1
Does the table represent a linear or nonlinear function? Explain.
a. b.
x 36912
y 40 32 24 16
As x increases by 3, y decreases
by 8. The rate of change is
constant. So, the function
is linear.
As x increases by 2, y increases
by different amounts. The rate
of change is not constant. So,
the function is nonlinear.
EXAMPLE
Identifying Functions from Graphs
2
2
Does the graph represent a linear or nonlinear function? Explain.
a.
x
y
1−2−323
−2
−1
1
2
3
−3
b.
x
y
1−2 −1−323
−1
1
2
3
−3
Does the table or graph represent a linear or nonlinear function?
Explain.
1.
xy
0
25
720
14 15
21 10
2.
xy
2 8
44
60
8 −4
3.
x
y
1−2−323
−1
2
3
−3
−2
The graph is not a line.
So, the function is nonlinear.
The graph is a line.
So, the function is linear.
Exercises 3–11
+3 +3 +3
−8 −8 −8
x 1357
y 2 113388
+2 +2 +2
+9 +22 +55
Key Vocabulary
nonlinear function,
p. 170
English
Spanish
Section 4.4 Comparing Linear and Nonlinear Functions 171
EXAMPLE
Standardized Test Practice
3
3
Which equation represents a nonlinear function?
A y = 4.7 B y =
π
x
C y =
4
—
x
D y = 4(x − 1)
The equations y = 4.7, y =
π
x, and y = 4(x − 1) can be rewritten in
slope-intercept form. So, they are linear functions.
The equation y =
4
—
x
cannot be rewritten in slope-intercept form.
So, it is a nonlinear function.
The correct answer is C .
EXAMPLE
Real-Life Application
4
4
Account A earns simple interest. Account B earns compound interest.
The table shows the balances for 5 years. Graph the data and compare
the graphs.
t
y
110
130
150
170
100
0
120
140
160
64253107
Year
Balance (dollars)
Account B
Savings Account
Account A
The balance of Account A has a constant rate of change of $10.
So, the function representing the balance of Account A is linear.
The balance of Account B increases by different amounts each year.
Because the rate of change is not constant, the function representing
the balance of Account B is nonlinear.
Does the equation represent a linear or nonlinear function? Explain.
4. y = x + 5 5. y =
4x
—
3
6. y = 1 − x
2
Remember
The simple interest
formula is given by
I = Prt.
●
I is the simple interest
●
P is the principal
●
r is the annual
interest rate
●
t is the time in years
Exercises 12–14
Year, t
Account A
Balance
Account B
Balance
0 $100 $100
1 $110 $110
2 $120 $121
3 $130 $133.10
4 $140 $146.41
5 $150 $161.05
English
Spanish
Exercises
4.4
172 Chapter 4 Functions
1. VOCABULARY Describe the difference between a linear function and a
nonlinear function.
2. WHICH ONE DOESN’T BELONG? Which equation does not belong with the
other thr
ee? Explain your reasoning.
5y = 2x
y =
2
—
5
x
10y = 4x
5xy = 2
9
+(-6)=3
3
+(-3)=
4
+(-9)=
9
+(-1)=
Graph the data in the table. Decide whether the function is linear or nonlinear.
3.
x 0123
y 4 8 12 16
4.
x 1234
y 12624
5.
x 6543
y 21 15 10 6
6.
x −1012
y −7 −31 5
Does the table or graph represent a linear or nonlinear function? Explain.
7.
x
y
3
4
2
−3
−4
−2
−1
4
21−2 −1−3−4
8.
x
y
3
4
2
1
−3
−4
−2
4
2 31−2 −1−3−4
9.
x 5 111723
y 7 111519
10.
x −3 −11 3
y 9119
11. VOLUME The table shows the volume V (in cubic feet) of a cube with a side length
of x feet. Does the table represent a linear or nonlinear function? Explain.
Side Length, x 12345678
Volume, V 1 8 27 64 125 216 343 512
Help with Homework
1
1
2
2
English
Spanish
Section 4.4 Comparing Linear and Nonlinear Functions 173
Classify the angle as acute, obtuse, right, or straight.
20.
21. 22. 23.
24. MULTIPLE CHOICE What is the value of x?
30°
x °
A 30 B 60 C 90 D 180
Does the equation represent a linear or nonlinear function? Explain.
12. 2x + 3y = 7 13. y + x = 4x + 5 14. y =
8
—
x
2
15. SUNFLOWER SEEDS The table shows the cost
y (in dollars) of x pounds of sunfl o
wer seeds.
a. What is the missing y-value that makes
the table represent a linear function?
b. Write a linear function that represents
the cost y of x pounds of seeds.
16. LIGHT The frequency y (in ter
ahertz) of a light wave is a function of its
wavelength x (in nanometers). Does the table represent a linear or nonlinear
function? Explain.
Color Red Yellow Green Blue Violet
Wavelength, x 660 595 530 465 400
Frequency, y 454 504 566 645 749
17. LIGHTHOUSES The table shows the
heights x (in feet) of four Florida
lighthouses and the number y of
steps in each. Does the table represent
a linear or nonlinear function? Explain.
18. PROJECT The wooden bars of a xylophone produce different
musical notes when str
uck. The pitch of a note is determined
by the length of the bar. Use the Internet or some other
reference to decide whether the pitch of a note is a linear
function of the length of the bar.
19.
The radius of the base of a cylinder is 3 feet. Is the volume of
the cylinder a linear or nonlinear function of the height of the cylinder?
Lighthouse Height, x Steps, y
Ponce de Leon Inlet 175 213
St. Augustine 167 219
Cape Canaveral 145 179
Key West 86 98
Pounds, x Cost, y
2 2.80
3?
4 5.60
3
3
Re
Re
d
d
Ye
ll
ll
ow
Gr
ee
n
Bl
Bl
ue
Vi
i
l
l
ol
et
English
Spanish
SKILLS REVIEW HANDBOOK
SECTION 1.1
