## Exponential Functions – Glencoe

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```Exponential Functions
Why?                                                                                             35
y

Tarantulas (hundreds)
30
Then                       Tarantulas can appear scary with their large hairy                                               25
You simplified numerical   bodies and legs, but they are harmless to humans.                                                20
expressions involving      The graph shows a tarantula spider population that                                               15
exponents. (Lesson 1-2)    increases over time. Notice that the graph is neither                                            10
5
Now                        The graph represents the function y = 3(2) x. This is an
-2-10          1 2 3 4 5 6x

Graph exponential        example of an exponential function.
functions.                                                                                                                    Years Since 2010
Identify data that
display exponential      Graph Exponential Functions An exponential function is a function of the form
behavior.                y = ab x, where a  0, b > 0, and b  1. Notice that the base is a constant and the
exponent is a variable. Exponential functions are nonlinear and nonquadratic
New Vocabulary             functions.
exponential function
For Your
Key Concept                   Exponential Function
Math Online
glencoe.com                  Words        An exponential function is a function that can be
Extra Examples                          described by an equation of the form y = ab x,
Personal Tutor                          where a  0, b > 0, and b  1.
Self-check Quiz                                                                                                                       x
y= _
1
Homework Help              Examples y = 2(3) x                        y = 4x                                                    (2)

EXAMPLE 1           Graph with a > 0 and b > 1
a. Graph y = 3 . Find the y-intercept, and state the domain and range.
x

y
x     3x         y
-2   3 -2        _
1
9
-1   3   -1      _
1
3
x
0     3   0
1                 y=3

1     31         3
2     32         9
0                                   x

Graph the ordered pairs, and connect the points with a smooth curve.
The graph crosses the y-axis at 1, so the y-intercept is 1.
The domain is all real numbers, and the range is all positive real numbers.
b. Use the graph to approximate the value of 3 0.7.
The graph represents all real values of x and their corresponding values
of y for y = 3 x. So, when x = 0.7, y is about 2. Use a calculator to confirm
this value: 3 0.7  2.157669.
1A. Graph y = 7 x. Find the y-intercept, and state the domain and range.
1B. Use the graph to approximate the value of y = 7 0.5 to the nearest tenth.
Use a calculator to confirm the value.
Personal Tutor glencoe.com

Lesson 9-6 Exponential Functions                                  567
The graphs of functions of the form y = ab x, where a > 0 and b > 1, all have the
same shape as the graph in Example 1. The greater the base or b-value, the faster the
graph rises as you move from left to right on the graph. The graphs of functions of
the form y = ab x, where a > 0 and 0 < b < 1, also have the same general shape.

StudyTip                        EXAMPLE 2             Graph with a > 0 and 0 < b < 1
a < 0 If the value of a                    1 x
is less than 0, the graph     a. Graph y = _   (3)
. Find the y-intercept, and state the domain and range.
will be reflected across
the x-axis.                                    x           (_
3)
1 x
y
y

1 -2
-2         (_
3)
9
x
y= _
1
(3)
0
0          (_
1
3)
1

2
2          (_
1
3)
_
1
9
0                x

The y-intercept is 1. The domain is all real numbers, and the range is all
positive real numbers. Notice that as x increases, the y-values decrease less
rapidly.
-1.5
b. Use the graph to approximate the value of _
1
(3)          .
When x = -1.5, the value of y is about 5. Use a calculator to confirm
this value:
KEYSTROKES:   ( 1  3
-1.5   ENTER   5.196152.

x
2A. Graph y = _
1
(2)
- 1. Find the y-intercept, and state the domain and range.
-2.5
2B. Use the graph to approximate the value of _
1
()           - 1 to the nearest tenth.
2
Use a calculator to confirm the value.
Personal Tutor glencoe.com

Exponential functions occur in many real world situations.

EXAMPLE 3             Use Exponential Functions to Solve Problems
SODA The consumption of soda has increased each year since 2000. The
function C = 179(1.029) t models the amount of soda consumed in the world,
where C is the amount consumed in billions of liters and t is the number of
years since 2000.
a. Graph the function. What values of C and t are
The United States is the             meaningful in the context of the problem?
largest soda consumer in
the world. In a recent year,         Since t represents time, t > 0. At t = 0, the
the United States                    consumption is 179 billion liters. Therefore, in the
accounted for one third of           context of this problem C > 179 is meaningful.
the world's total soda
consumption.
Source: Worldwatch Institute                                                                                      [-50, 50] scl: 10 by [0, 350] scl: 25

568 Chapter 9 Quadratic and Exponential Functions
b. How much soda was consumed in 2005?
C = 179(1.029) t  Original equation
= 179(1.029) 5  t=5
= 206.5         Use a calculator.
The world soda consumtion in 2005 was approximately 206.5 billion liters.

3. A certain bacteria doubles every 20 minutes. Beginning with 10 cells in a
culture, the population can be represented by the function B = 10(2) t, where B
is the number of bacteria cells and t is the time in 20 minute increments. How
many will there be after 2 hours?
Personal Tutor glencoe.com

Identify Exponential Behavior Recall from Lesson 3-3 that linear functions have a
constant rate of change. Exponential functions do not have constant rates of change,
but they do have constant ratios.

EXAMPLE 4          Identify Exponential Behavior
Determine whether the set of data shown below displays exponential behavior.
Write yes or no. Explain why or why not.
Problem-SolvingTip
x       0         5         10        15              20        25
Make an Organized
y           64        32        16            8            4        2
List Making an
organized list of
x-values and
Method 1   Look for a pattern.
corresponding y-values        The domain values are at regular intervals of 5. Look for a common factor among
is helpful in graphing        the range values.
the function. It can also
patterns in the data.
_ _ _ _ _
1 1 1 1 1
2        2    2         2           2

The range values differ by the common factor of _
1
.
2
Since the domain values are at regular intervals and the range values differ
by a positive common factor, the data are probably exponential. Its equation
x
may involve _
1
(2)
.

Method 2   Graph the data.                                                                                        y
64
Plot the points and connect them with a smooth curve.                                                        56
48
The graph shows a rapidly decreasing value of y as x                                                         40
increases. This is a characteristic of exponential                                                           32
behavior in which the base is between 0 and 1.                                                               24
StudyTip                                                                                                                                    16
8
The graph of an              Check Your Progress                                                                                         -50        5 10 15 20 25 30 35x

exponential function
may resemble part of          4. Determine whether the set of data shown below displays exponential behavior.
the graph of a                   Write yes or no. Explain why or why not.
sure to check for a                                                x        0         3         6            9            12        15
pattern as well as to                                              y        12        16        20           24           28        32
look at a graph.
Personal Tutor glencoe.com

Lesson 9-6 Exponential Functions                             569
Examples 1 and 2        Graph each function. Find the y-intercept, and state the domain and range. Then
pp. 567568   use the graph to determine the approximate value of the given expression. Use a
calculator to confirm the value.
1. y = 2 x; 2 1.5                                          2. y = -5 x; -5 0.5
x         -0.5                                     1 x              0.5
3. y = - _
1
;- _
(5) (5)
1
4. y = 3 _   ;3 _
(4) (4)
1

Graph each function. Find the y-intercept, and state the domain and range.
5. f(x) = 6 x + 3                                          6. f(x) = 2 - 2 x

Example 3        7. BIOLOGY The function f(t) = 100(1.05) t models the growth of a fruit fly
pp. 568569       population, where f(t) is the number of flies and t is time in days.
a. What values for the domain and range are reasonable in the context of this
situation? Explain.
b. After two weeks, approximately how many flies are in this population?

Example 4       Determine whether the set of data shown below displays exponential behavior.
p. 569   Write yes or no. Explain why or why not.
8.    x    1         2     3      4        5        6
9.   x         2    4      6       8         10          12
y   -4         -2    0      2        4        6           y         1    4      16      64        256     1024

= Step-by-Step Solutions begin on page R12.
Practice and Problem Solving                                                                           Extra Practice begins on page 815.

Examples 1 and 2        Graph each function. Find the y-intercept, and state the domain and range. Then
pp. 567568   use the graph to determine the approximate value of the given expression. Use a
calculator to confirm the value.
1 x           1.5                1 x _                  0.5
10. y = 2  8 x, 2(8) -0.5                 11. y = 2  _   ;2 _
(6) (6)
1
12. y = _   ( 12 ) ( 12 )
; 1
13. y = -3  9 x, -3(9) -0.5               14. y = -4  10 x, -4(10) -0.5 15. y = 3  11 x, 3(11) -0.2
Graph each function. Find the y-intercept, and state the domain and range.
16. y = 4 x + 3                 17. y = _
1( x
2 - 8)           18. y = 5(3 x) + 1               19. y = -2(3 x) + 5
2
Example 3       20. BIOLOGY A population of bacteria in a culture increases according to the model
pp. 568569       p = 300(2.7) 0.02t, where t is the number of hours and t = 0 corresponds to 9:00 a.m.
a. Use this model to estimate the number of bacteria at 11 a.m.
b. Graph the function and name the p-intercept. Describe what the p-intercept
represents, and describe a reasonable domain and range for this situation.

Example 4       Determine whether the set of data shown below displays exponential behavior.
p. 569   Write yes or no. Explain why or why not.
21     x   -4         0     4       8           12         22.   x         -6   -3     0       3
y    2         -4    8      -16          32               y         5    10     15      20

23.    x   -8         -6    -4     -2                      24.   x         20   30     40           50          60
y   0.25       0.5   1          2                         y         1    0.4   0.16      0.064          0.0256

570 Chapter 9 Quadratic and Exponential Functions
25 PHOTOGRAPHY Jameka is enlarging a photograph to make a poster for school. She
will enlarge the picture repeatedly at 150%. The function P = 1.5 x models the
B new size of the picture being enlarged, where x is the number of enlargements.
How big is the picture after it has been enlarged 4 times?

26. FINANCIAL LITERACY Daniel invested \$500 into a savings account. The equation
A = 500(1.005) 12t models the value of Daniel's investment A after t years. How
much will Daniel's investment be worth in 8 years?

Identify each function as linear, quadratic, or exponential.
The world's largest
27.               y          28.              y             29.          y
photograph, named
The Great Picture, was
created by a group of
photographers known as
The Legacy Project. The                                                                                 0           x
photograph has an area of                                               0              x
3375 square feet.
Source: Photoshop Support
0          x

30. y = 4 x + 3              31. y = 2x(x - 1)              32. 5x + y = 8

33. GRADUATION The number of graduates at a high school has increased by a factor
of 1.055 every year since 2001. In 2001, 110 students graduated. The function
N = 110(1.055) t models. N, the number of students expected to graduate t year
after 2001. How many students will graduate in 2012?

C Describe the graph of each equation as a transformation of the graph of y = 2 x.
34. y = 2 x + 6              35. y = 3(2) x                 36. y = -_
1 ( )x
2
4
x
37. y = -3 + 2    x
38. y = _
()
1
39. y = -5(2) x
2

40. DEER The deer population at a national park doubles every year. In 2000, there
were 25 deer in the park. The function N = 25(2) t models the number of deer N
in the national park t years after 2000. What will the deer population in the park
be in 2015?

H.O.T. Problems             Use Higher-Order Thinking Skills

41. CHALLENGE Write an exponential function that passes through the points at (0, 3)
and (1, 6).

42. REASONING Determine whether the graph of y = ab x, where a  0, b > 0, and
b  1, sometimes, always, or never has an x-intercept. Explain your reasoning.

43. OPEN ENDED Choose an exponential function that represents a real-world
situation, and graph the function. Analyze the graph.

44. REASONING Compare and contrast an exponential function of the form
y = ab x + c, where a  0, b > 0, and b  1 and a quadratic function of the
form y = ax 2 + c.

45. WRITING IN MATH Explain how to determine whether a set of data displays
exponential behavior.

Lesson 9-6 Exponential Functions   571```

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