exponential functions - ms. juengel's...

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

3.1 Exponential Functions

2

Quick Review

Evaluate the expression without using a calculator.

1. -1253

2.2764

3

3. 274/3

Rewrite the expression using a single positive exponent.

4. a-3( )2

Use a calculator to evaluate the expression.

5. 3.712935

3

Quick Review Solutions

( )6

3

3

4 / 3

2-3

Evaluate the expression without using a calculator.

1. -125

272. 64

3. 27 Rewrite the expression using a single positive e

-5

3481

1xponent.

4.

Use a calculator to evaa

a

5

luate the expression.

5. 3.71293 1.3

4

Today’s Objectives

Content Objective: n  Exponential Functions and Their Graphs n  The Natural Base e n  Exponential functions model many growth patterns,

including the growth of human and animal populations.

Language Objective: n  Use sentence frames to verbally determine a solution

to all turn and talk questions. Report your conclusions about exponential functions to the teacher.

5

Exponential Functions

Let a and b be real number constants.

An exponential function in x is a function that can be written in the form f (x) = a ⋅bx , where a is nonzero, b is positive, and b ≠ 1. The constant a is the initial value of f (the value at x = 0), and b is the base.

The domain/input of exponential functions are exponents. The range/output is the base, b, multiplied by itself [insert exponent here] times and vertically stretched or shrunk by a factor of a.

6

Finding an Exponential Function from its Table of Values

Determine formulas for the exponential function and whose values are given in the table below.

g h

7

Because g is exponential, g(x) = a ⋅bx .

Because g(0) = 4, a = 4.

Because g(1) = 4 ⋅b1 = 12, the base b = 3. So, g(x) = 4 ⋅3x.

8

Because h is exponential, h(x) = a ⋅bx .

Because h(0) = 8, a = 8.

Because h(1) = 8 ⋅b1 = 2, the base b = 1/ 4. So, h(x) = 8 ⋅ 14

⎛⎝⎜

⎞⎠⎟

x

.

9

Exponential Growth and Decay

For any exponential function f (x) = a ⋅bx and any real number x,f (x +1) = b ⋅ f (x).

If a > 0 and b >1, the function f is increasing and is an exponentialgrowth function. The base b is its growth factor.

If a > 0 and b <1, the function f is decreasing and is an exponentialdecay function. The base b is its decay factor.

10

Example Transforming Exponential Functions

Describe how to transform the graph of f (x) = 2x into the graph of g(x) = 2x-2.

11


-2Describe how to transform the graph of ( ) 2 into the graph of ( ) 2 .x xf x g x= =

The graph of g(x) = 2x-2 is obtained by translating the graph of f (x) = 2x by 2 units to the right.

The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.

12

Reflections: Transforming Exponential Functions

Describe how to transform the graph of f (x) = 2x into the graph of g(x) = 2-x.

13

Reflections: Transforming Exponential Functions

Describe how to transform the graph of f (x) = 2x into the graph of g(x) = 2-x.

Since the transformation is applied to x it will effect a horizontal property of the graph, therefore the graph of g(x) = 2−x is obtained by making a horizontal reflection of the graph. So flip f (x) = 2x across the y-axis to produce g(x) = 2−x.

14

The Natural Base e

e = lim

x→∞1+ 1

x⎛⎝⎜

⎞⎠⎟

x

Turn and Talk: •  When have we used limits? In your experience, what do

limits tell us?

•  Think about polynomial and rational functions what type of limit is indicated when x approaches infinity?

•  Find the limit numerically, and make a conjecture about the value of Euler’s constant also called the natural base e.

Report to me: •  Each group must state there answer to the teacher.

15

Exponential Functions and the Base e

Any exponential function f (x) = a ⋅bx can be rewritten as f (x) = a ⋅ekx , for any appropriately chosen real number constant k.

If a > 0 and k > 0, f (x) = a ⋅ekx is an exponential growth function.

If a > 0 and k < 0, f (x) = a ⋅ekx is an exponential decay function.

Turn and Talk: What is the relationship between b, k, and e? Justify your response using the properties of exponents. Report to me: •  Each group must state

there answer to the teacher.

16

Exponential Functions and the Base e

Turn and Talk: •  Classify these functions. •  What will they always have in common? •  Use the properties of exponents to explain why? Report to me: •  Each group must state there answer to the teacher.

17


Describe how to transform the graph of f (x) = ex into the graph of g(x) = e3x .

Since the transformation is applied to x it will effect a horizontal property of the graph, therefore the graph of g(x) = e3x is obtained by making a horizontal reflection of the graph. So _________ f (x) __________ by ___________ g(x) = e3x .

AM: Graph exponential functions

Slide 2- 18

1

LO: I know by looking at the equation that I have an a = ____>____, and a b =____<______. This means I have an example of exponential ___________ with a y-intercept of ________. Therefore, my answer is _____.

19

Graph Exponential Functions

2

LO: I know by looking at the equation that a = _____>_____, and b = _____<______value. This means I have an example of exponential ___________ with a y-intercept of ________. I also have a ________________shift of ____ units _____ and a ________ shift of ______ units ______. Therefore, my answer is ______. Report to me: •  Each group must state the answer to the teacher. •  Pick up your AM Practice on Objectives #1 and #2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

3.2 Exponential and Logistic Modeling

21

Quick Review

2

Convert the percent to decimal form or the decimal into a percent.1. 16%2. 0.053. Show how to increase 25 by 8% using a single multiplication.Solve the equation algebraically.4. 20 720Solve the equ

b⋅ =

3

ation numerically.5. 123 7.872b⋅ =

22

Quick Review Solutions

Convert the percent to decimal form or the decimal into a percent.1. 16% 2. 0.05 3. Show how to increase 25 by 8% using a single multiplication.Solve the equation algebraically.

0.165%

25 1 4

.08⋅2

3

. 20 720 Solve the equation numerically.5. 123 7.872

6

0. 4

b

b

⋅

⋅

±=

=

23

What you’ll learn about

n  Constant Percentage Rate and Exponential Functions n  Exponential Growth and Decay Models n  Using Regression to Model Population n  Other Logistic Models … and why Exponential functions model many types of unrestricted growth; logistic functions model restricted growth, including the spread of disease and the spread of rumors.

24

Constant Percentage Rate

Suppose that a population is changing at a constant percentage rate r, where r is the percent rate of change expressed in decimal form. Then the population follows the pattern shown.

0

0 0 0

Time in years Population0 (0) initial population1 (1) (1 )2 (2) (1)

P PP P Pr P rP P

= =

= + = +

= 2

0

3

0

0

(1 ) (1 )3 (3) (2) (1 ) (1 ) ( ) (1 ) t

r P rP P r P r

t P t P r

⋅ + = +

= ⋅ + = +

= +

M M

25

Exponential Population Model

0 0

If a population is changing at a constant percentage rate each year, then( ) (1 ) , where is the initial population, is expressed as a decimal,

and is time in years.

t

P rP t P r P r

t= +

26

Example Finding Growth and Decay Rates

Tell whether the population model ( ) 786,543 1.021 is an exponentialgrowth function or exponential decay function, and find the constant percentrate of growth.

tP t = ⋅

27

Example Finding Growth and Decay Rates

Tell whether the population model ( ) 786,543 1.021 is an exponentialgrowth function or exponential decay function, and find the constant percentrate of growth.

tP t = ⋅

Because 1 1.021, .021 0. So, is an exponential growth functionwith a growth rate of 2.1%.

r r P+ = = >

28

Example Finding an Exponential Function

Determine the exponential function with initial value=10, increasing at a rate of 5% per year.

29

Example Finding an Exponential Function

Determine the exponential function with initial value=10, increasing at a rate of 5% per year.

0Because 10 and 5% 0.05, the function is ( ) 10(1 0.05) or

( ) 10(1.05) .

t

t

P r P tP t

= = = = +

=

WP: Exponential Functions

Slide 2- 30

3

LO: We know that the initial value corresponds to a, so a = ______. Since the rate of growth is _______, the base will be equal to _____. This means our equation will be _______. Since time, x = ______, we can evaluate the function for x = ______. This give us a value of ________.


Slide 2- 31

3

LO: We know that the initial value corresponds to a, so a = ______. Since the rate of _______ is _______, the _____ will be equal to _____. This means our equation will be _______. Since time, x = ______, we can __________ the function for x = ______. This give us a value of ________.


Slide 2- 32

3

LO: We know that the initial value corresponds to a, so a = ______. Since the rate of _______ is _______, the _____ will be equal to _____. This means our equation will be _______. Since time, x = ______, we can __________ the function for x = ______. This give us a value of ________.

33

Example Modeling Bacteria Growth

Suppose a culture of 200 bacteria is put into a petri dish and the culturedoubles every hour. Predict when the number of bacteria will be 350,000.

34

Example Modeling Bacteria Growth

Suppose a culture of 200 bacteria is put into a petri dish and the culturedoubles every hour. Predict when the number of bacteria will be 350,000.

2

400 200 2800 200 2

( ) 200 2 represents the bacteria population hr after it is placedin the petri dish. To find out when the population will reach 350,000, solve350,000 200 2 for using

t

t

P t t

t

= ⋅

= ⋅

= ⋅

= ⋅

M

a calculator.10.77 or about 10 hours and 46 minutes.t =

35

Example Modeling U.S. Population Using Exponential Regression

Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

36

Example Modeling U.S. Population Using Exponential Regression

Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

103

Let ( ) be the population (in millions) of the U.S. years after 1900.Using exponential regression, find a model ( ) 80.5514 1.01289 .To find the population in 2003 find (103) 80.5514 1.01289 3

t

P t tP t

P= ⋅

= ⋅ ≈ 01.3.

37

Maximum Sustainable Population

Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population.

38

Example Modeling a Rumor

-0.9

A high school has 1500 students. 5 students start a rumor, which spreads logistically so that ( ) 1500 /(1 29 ) models the number of studentswho have heard the rumor by the end of days, where

tS t et

= + ⋅

0 is the day therumor begins to spread.(a) How many students have heard the rumor by the end of Day 0?(b) How long does it take for 1000 students to hear the rumor?

t =

39

Example Modeling a Rumor

-0.9

A high school has 1500 students. 5 students start a rumor, which spreads logistically so that ( ) 1500 /(1 29 ) models the number of studentswho have heard the rumor by the end of days, where

tS t et

= + ⋅

0 is the day therumor begins to spread.(a) How many students have heard the rumor by the end of Day 0?(b) How long does it take for 1000 students to hear the rumor?

t =

-0.9 ( 0 )(a) (0) 1500 /(1 29 ) 1500 /(1 29 1) 1500 / 30 50. So 50 students have heard the rumor by the end of day 0.

S e= + ⋅

= + ⋅

= =

-0.9(b) Solve 1000 1500 /(1 29 ) for .4.5. So 1000 students have heard the rumor half way

through the fifth day.

te tt

= + ⋅

≈

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