7-1
Attributes of Exponential Functions
TEKS FOCUS
VOCABULARY
TEKS (2)(A) Graph the functions
3
f(x) = 1x, f(x) = 1x , f(x) = x3 , f(x) = 2x
,
x
f(x) = b , f(x) = 0 x 0 , and f(x) = logb(x)
where b is 2, 10, and e, and, when
applicable, analyze the key attributes such
as domain, range, intercepts, symmetries,
asymptotic behavior, and maximum and
minimum given an interval.
TEKS (1)(A) Apply mathematics to
problems arising in everyday life, society,
and the workplace.
Additional TEKS (1)(C), (1)(E), (5)(B),
(5)(D), (7)(I), (8)(C)
ĚAsymptote – a line that a graph
approaches as x or y increases in
absolute value.
ĚDecay factor – In an exponential
ĚExponential growth – See growth
factor.
ĚGrowth factor – In an exponential
growth function y = abx , with
a 7 0 and b 7 1, the value b is the
growth factor.
abx ,
decay function y =
with
a 7 0 and 0 6 b 6 1, the value b is
the decay factor.
ĚApply – use knowledge or
ĚExponential decay – See decay
information for a specific
purpose, such as solving a
problem
factor.
ĚExponential function – a function
with the general form y = abx ,
a ≠ 0, with b 7 0, and b ≠ 1.
ESSENTIAL UNDERSTANDING
You can represent repeated multiplication with a function of the form y = abx where
b is a positive number other than 1.
Concept Summary Exponential Functions
For the function y = abx, the y-intercept is (0, a), the
domain is all real numbers, the asymptote is y = 0,
and the range is y 7 0.
Exponential growth
a 7 0 and b 7 1
Exponential decay
a 7 0 and 0 6 b 6 1
y
Exponential
Growth
Exponential
Decay
x
O
The x-axis is an asymptote.
Key Concept
Exponential Growth and Decay
For growth or decay to be exponential, a quantity
changes by a fixed percentage each time period.
You can model exponential growth or decay with
the function shown at the right.
Amount after t
time periods
A(t) = a(1 + r)t
Initial amount
266
Lesson 7-1
Attributes of Exponential Functions
Rate of growth (r 7 0)
or decay (r 6 0)
Number of time periods
Problem 1
P
Comparing the Graphs of y = 2x and y = 10 x
A Graph y = 2x . State the domain and range.
Step 1 Make a table of values.
How does making a
table help you sketch
the graph?
The table shows
coordinates of several
points on the graph.
Step 2 Plot and connect the points.
x
2x
y
x
2x
y
8
4
24
0
20
1
6
3
23
1
21
2
4
2
22
2
22
4
2
1
21
1
16 0.0625
1
8 0.125
1
4 0.25
1
2 0.5
3
23
8
4
2
y
x
O
2
4
The domain of y = 2x is all real numbers, or ( - ∞, ∞ ). The range is y 7 0,
or (0, ∞ ).
B Graph y = 10x . State the domain and range.
Step 1 Make a table of values.
Step 2 Plot and connect the points.
x
10x
y
-2
10 -2
-1
10 -1
1
100
1
10
0
10 0
1
4
1
101
10
2
0
102
100
10
y
8
6
-4
-2
O
x
2
The domain of y = 10 x is all real numbers, or ( - ∞, ∞ ). The range is y 7 0,
or (0, ∞ ).
C Compare the graphs above. Describe the intercepts and the asymptotes.
Both graphs are increasing for all values of x, but y = 10 x increases more rapidly
than y = 2x . Both graphs have a y-intercept of 1, and no x-intercept. Since the
graphs approach the x-axis as x decreases, y = 0 is a horizontal asymptote of
each graph.
PearsonTEXAS.com
267
Problem 2
P
Analyzing Attributes of the Graphs of y = 2 x and y = 10 x
On separate grids, graph the functions f (x) = 2x and g (x) = 10 x . Use the graphs to
answer the following questions.
What are the minimum and maximum of f (x) on the interval [ −2, 1]? What
are the minimum and maximum of g (x) on the same interval?
From the graphs, it is clear that f and g increase as x increases.
f (x)
How do you evaluate
the functions for
negative values of x?
2-x and 10-x can be
written as 21x and 101 x .
8
−4
−2
g(x)
y
10
6
8
4
6
2
4
O
x
2
2
4
−2
The minimum value of f (x) on the
interval is f ( -2) = 2-2 = 14 . The
maximum value of f (x) on the interval
is f (1) = 21 = 2.
y
O
x
2
4
The minimum value of g (x) on
1
the interval is g ( -2) = 10-2 = 100
.
The maximum value of g (x) on the
interval is g (1) = 101 = 10.
Problem
bl
3
TEKS Process Standard (1)(A)
Identifying Exponential Growth and Decay
Identify each function or situation as an example of exponential growth
or decay. What is the y-intercept?
A y = 12(0.95)x
What quantity does
the y-intercept
represent?
The y-intercept is the
amount of money at
t = 0, which is the
initial investment.
268
Lesson 7-1
Since 0 6 b 6 1, the function
represents exponential decay.
The y-intercept is (0, a) = (0, 12).
B y = 0.25(2)x
Since b 7 1, the function
represents exponential growth.
The y-intercept is (0, a) = (0, 0.25).
C You put $1000 into a college savings account for four years. The account
pays 5% interest annually.
The amount of money in the bank grows by 5% annually. It represents
exponential growth. The y-intercept is 1000, which is the dollar value of the
initial investment.
Attributes of Exponential Functions
Problem 4
P
TEKS Process Standard (1)(A)
Modeling Exponential Growth
You invested $1000 in a savings account at the end of 6th grade. The account pays
5% annual interest. How much money will be in the account after six years?
Step 1 Determine if an exponential function is a reasonable model.
The money grows at a fixed rate of 5% per year. An exponential model,
A (t) = a (1 + r)t , is appropriate.
Step 2 Define the variables.
Let t = the number of years since the money was invested.
Let A (t) = the amount in the account after each year.
What is the growth
rate r?
It is the annual interest
rate, written as a
decimal: 5% = 0.05.
Step 3 Use the model to solve the problem.
S
A (6) = 1000(1 + 0.05)6
=
1000(1.05)6
Substitute a = 1000, r = 0.05, and t = 6.
Simplify.
≈ $1340.10
The account contains $1340.10 after six years.
Problem
bl
5
TEKS Process Standard (1)(E)
Using Exponential Growth
Suppose you invest $1000 in a savings account that pays 5% annual interest. If you
make no additional deposits or withdrawals, how many years will it take for the
account to grow to at least $1500?
How can you make
a table to solve
this problem?
Define the variables,
write an equation, and
enter it into a graphing
calculator. Then you can
inspect a table to find
the solution.
Define the variables.
Determine the model.
Make a table using
the table feature on a
graphing calculator. Find
the input when the output
is 1500.
Let t = the number of years.
Let A (t) = the amount in the account after t years.
A (t) = 1000(1 + 0.05)t
= 1000(1.05)t
X
Y1
4
5
6
7
8
9
10
1215.5
1276.3
1340.1
1407.1
1477.5
1551.3
1628.9
Y1=1551.32821598
The account pays interest
only once a year. The
balance after the 8th
year is not yet $1500.
The account will not contain $1500 until
the ninth year. After nine years, the balance
will be $1551.33.
PearsonTEXAS.com
269
Problem 6
P
Writing an Exponential Function
STEM
Endangered Species The table shows the world
population of the Iberian lynx in 2003 and 2004.
If this trend continued and the population
decreased exponentially, how many Iberian
lynx were there in 2014?
Use the general form of the exponential equation,
y = abx = a(1 + r)x.
Step 1 Define the variables.
How can you find the
value of r?
You can use the
populations for two
consecutive years to
find r.
Let x = the number of years since 2003.
Let y = the population of the Iberian lynx.
Step 2 Determine r.
S
Use the populations for 2003 and 2004.
r=
=
y2 - y1
y1
World Population
of Iberian Lynx
120 - 150
150
Year
= -0.2
Population
Step 3 Use r to determine b.
b = 1 + r = 1 + ( -0.2) = 0.8
Step 4 Write the model.
y = abx
150 = a (0.8)0
150 = a
How do you find the
x-value corresponding
to 2014?
The initial x-value
corresponds to 2003, so
find the difference.
Solve for a using the initial
values x 0 and y 150.
The model is y = 150(0.8)x.
Step 5 Use the model to find the population in 2014.
S
For the year 2014, x = 2014 - 2003 = 11.
y = 150(0.8)x
= 150(0.8)11
≈ 13
If the 2003–2004 trend continued, there were approximately 13 Iberian
lynx in the wild in 2014.
270
Lesson 7-1
Attributes of Exponential Functions
2003 2004
150
120
HO
ME
RK
O
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Determine if the statement is sometimes, always, or never true.
1. For a given x-value, the graph of y = 10x lies above the graph of y = 2x .
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completing your homework,
go to PearsonTEXAS.com.
2. For a given x-value, the y-value of y = 2x is positive.
3. The domain of an exponential function y = bx , where b 7 1, is all real numbers.
4. Graph the functions y = 2x , y = 5x , and y = 10x on a graphing calculator.
Then make a conjecture about the graph of y = bx , where b 7 1.
For each exponential function, sketch the graph on a coordinate plane. Then
identify any asymptotes and find the minimum and maximum values of f (x)
on the given interval.
5. f (x) = 2x; [ -2, 4]
6. f (x) = 10 x; [ -7, 5]
7. Explain how the asymptotic behavior of the graph of f (x) = 2x affects the x- and
y-intercepts of the graph.
8. A student claims the line x = 4 is a vertical asymptote of the graph of f (x) = 10x
because the graph appears to become vertical as x approaches 4. Is the student
correct? Construct a graph as evidence of your position and explain your reasoning.
Without graphing, determine whether the function represents exponential
growth or exponential decay. Then find the y-intercept.
9. y = 129(1.63)x
10. f (x) = 2(0.65)x
17 x
( )
11. y = 12 10
x
12. y = 0.8 18
()
13. Apply Mathematics (1)(A) Suppose you deposit $2000 in a savings account
that pays interest at an annual rate of 4%. If no money is added or withdrawn
from the account, answer the following questions.
a. How much will be in the account after 3 years? after 18 years?
b. How many years will it take for the account to contain $3000?
14. Apply Mathematics (1)(A) The function y = 20(0.975)x models the intensity of
sunlight beneath the surface of the ocean. The output y represents the percent
of surface sunlight intensity that reaches a depth of x feet. The model is accurate
from about 20 feet to about 600 feet beneath the surface.
a. Find the percent of sunlight 50 feet beneath the surface of the ocean.
b. Find the percent of sunlight at a depth of 370 feet.
p
15. Apply Mathematics (1)(A) Determine which situation best
matches the graph.
A. A population of 120 cougars decreases 98.75% yearly.
75
B. A population of 120 cougars increases 1.25% yearly.
C. A population of 115 cougars decreases 1.25% yearly.
D. A population of 115 cougars decreases 50% yearly.
25
O
t
10
20
30
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16. Apply Mathematics (1)(A) The population of a certain animal species
decreases at a rate of 3.5% per year. You have counted 80 of the animals in the
habitat you are studying.
a. Write a function that models the change in the animal population.
b. Select Tools to Solve Problems (1)(C) Graph the function. Estimate the
number of years until the population first drops below 15 animals.
17. Apply Mathematics (1)(A) While you are waiting for
your tennis partner to show up, you drop your tennis
ball from 5 feet. Its rebound was approximately
35 inches on the first bounce and 21.5 inches on the
second. What exponential function would be a good
model for the bouncing ball?
35 in.
18. Apply Mathematics (1)(A) The value of an industrial
machine has a decay factor of 0.75 per year. After
six years, the machine is worth $7500. What was the
original value of the machine?
5 ft.
21.5 in.
19. Create Representations to Communicate
Mathematical Ideas (1)(E) Write a problem that
could be modeled with y = 20(1.1)x.
Write an exponential function to model each situation. Find each amount
after the specified time.
20. A population of 120,000 grows 1.2% per year for 15 years.
21. A population of 1,860,000 decreases 1.5% each year for 12 years.
TEXAS Test Practice
T
22. Which function represents the value after x years of a new delivery van that costs
$25,000 and depreciates 15% each year?
A. y = -15(25,000)x
C. y = 25,000(0.85)x
B. y = 25,000(0.15)x
D. y = 25,000(1.15)x
23. Which graph represents the equation y = x2 - x - 2?
y
y
y
F.
H.
G.
2
2
2
O
2x
2
O
x
O
J.
2x
2
2x
2
2
272
Lesson 7-1
Attributes of Exponential Functions
y