Exponential Functions – Glencoe

Text Preview:
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
                           linear nor quadratic.
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.
                            Check Your Progress
                             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.

                                 Check Your Progress
                                                          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.

                              Check Your Progress
                               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
 help you identify                                          64 32 16 8 4 2
 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
 Checking Answers
 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.
 quadratic function. Be
 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
  Check Your Understanding
  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
Download Link:
Share Link: Forum Link:

More on Science & Technology

  • Picture: Math Boxes - Everyday Math - Login

    Math Boxes – Everyday Math – Login

    File Size: 1,819.53 KB, Pages: 5, Views: 1,034,172 views

    Math Boxes Objectives To introduce My Reference Book; and to introduce the t Math Boxes routine. www.everydaymathonline.com ePresentations eToolkit Algorithms EM Facts Family Assessment Common Curriculum Interactive Practice Workshop Letters Management Core State Focal Points Teacher's GameTM Standards Lesson Guide Teaching the Lesson Ongoing Learning …
  • Picture: A Study of the Relationship Between Students Anxiety and

    A Study of the Relationship Between Students Anxiety and

    File Size: 72.91 KB, Pages: 7, Views: 1,002,907 views

    US-China Education Review B 4 (2011) 579-585 Earlier title: US-China Education Review, ISSN 1548-6613 A Study of the Relationship Between Students' Anxiety and Test Performance on State-Mandated Assessments Rosalinda Hernandez, Velma Menchaca, Jeffery Huerta University of Texas Pan American, Edinburg, USA This study examined whether …
  • Picture: HIGH-EFFICIENCY UPFLOW FURNACE INSTALLER'S  - Crown Boiler

    HIGH-EFFICIENCY UPFLOW FURNACE INSTALLER’S – Crown Boiler

    File Size: 534.22 KB, Pages: 27, Views: 994,383 views

    HIGH-EFFICIENCY UPFLOW FURNACE INSTALLER'S INFORMATION MANUAL D ES IG N CE R TI F I ED ATTENTION, INSTALLER! After installing the ATTENTION, USER! Your furnace installer should furnace, show the user how to turn off gas and electricity to give you the documents listed on …
  • Picture: Raven/Johnson Biology 8e Chapter 12 1.

    Raven/Johnson Biology 8e Chapter 12 1.

    File Size: 99.62 KB, Pages: 9, Views: 79,495 views

    Raven/Johnson Biology 8e Chapter 12 1. A true-breeding plant is one that-- a. produces offspring that are different from the parent b. forms hybrid offspring through cross-pollination c. produces offspring that are always the same as the parent d. can only reproduce with itself The …
  • Picture: Math Skills for Business- Full Chapters 1 U1-Full Chapter

    Math Skills for Business- Full Chapters 1 U1-Full Chapter

    File Size: 3,860.88 KB, Pages: 188, Views: 95,200 views

    Math Skills for Business- Full Chapters 1 U1-Full Chapter- Algebra Chapter3 Introduction to Algebra 3.1 What is Algebra? Algebra is generalized arithmetic operations that use letters of the alphabet to represent known or unknown quantities. We can use y to represent a company's profit or …

Leave a Reply

Your email address will not be published. Required fields are marked *