Chapter 3 Exponential and Logarithmic Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 3.1 Exponential Functions

Chapter 3Exponential andLogarithmic Functions

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

3.1 Exponential Functions


• Evaluate exponential functions.• Graph exponential functions.• Evaluate functions with base e.• Use compound interest formulas.

Objectives:


What is an exponential function?What does an exponential function look like?

BaseBase

ExponentExponentand and

Independent Independent VariableVariableJust some Just some

number number that’s not 0that’s not 0

Why not 0?Why not 0?

Dependent Dependent VariableVariable

Obviously, it must have something to do with an exponent!


The Basis of Bases

The base carries the meaning of the function.

1) determines exponential growth or decay.

2)base is a positive number; however, it cannot be 1.


Let’s examine exponential functions. They are different than any of the other types of functions we’ve studied because the independent variable is in the exponent.

xxf 2

Let’s look at the graph of this function by plotting some points. x 2x

3 8 2 4 1 2 0 1

-1 1/2 -2 1/4 -3 1/8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

2

121 1 f

Recall what a negative exponent means:

BASE


Using graphing calculator plot (graph) :

2

3

4

x

x

x

y

y

y

xxf 3

xxf 4

xxf 2

Compare graphs in your groups:

1)Similarities

2)Differences


xxf 2

xxf 3

Similarity of Graphs of Exponential Function where a > 1

xaxf

What is the domain of an exponential function?

1. Domain is all real numbers

xxf 4

What is the range of an exponential function?

2. Range is positive real numbers

What is the x intercept of these exponential functions?

3. There are no x intercepts because there is no x value that you can put in the function to make it = 0

What is the y intercept of these exponential functions?

4. The y intercept is always (0,1) because a 0 = 1

5. The graph is always increasing

Are these exponential functions increasing or decreasing?

6. A horizontal asymptote y = 0 (x-axis)

Can you see the horizontal asymptote for these functions?

What is different?

Steepness


Graphing Exponential functions by Transformations

What do you remember about Transformations of any functions: ---- shifts, reflections, stretching, compressingCoefficients/constants “a”, “h” and “k”

On your graphing calculators : 2

2 3

2 4

x

x

x

y

y

y


Transformations

Shifts the graph up if k > 0

Shifts the graph down if k < 0

2xy

2 3xy

2 4xy Vertical translation f(x) = bx + k


Shifts to the left if h > 0

Shifts to the right if h < 0

2xy

( 3)2 xy

( 4)2 xy

On your graphing calculators :

3

4

2

2

2

x

x

x

y

y

y

Horizontal translation: f(x)=bx-h


Reflecting

reflects over the x-axis.

reflects over the y-axis.

2xy

2xy

2 xy

On your graphing calculators :

1

2

1 2

2

x

x

x

y

y

y

( ) 1 xf x b

1( ) xf x b


Stretches the graph if a > 1

Shrinks the graph if 0 < a < 1

2xy

4(2 )xy 1

(2 )4

xy

Vertical stretching or shrinking, f(x)=abx:

On your graphing calculators : 2

4 2

12

4

x

x

x

y

y

y


Foldable: Graphing Exponential Functions

13


Let’s take a second look at the base of an exponential function.(It can be helpful to think about the base as the object that is being multiplied by itself repeatedly.)

Why can’t the base be negative?

Why can’t the base be zero?

Why can’t the base be one?



Example: Evaluating an Exponential Function

The exponential function models the average amount spent, f(x), in dollars, at a shopping mall after x hours. What is the average amount spent, to the nearest dollar, after three hours at a shopping mall?

We substitute 3 for x and evaluate the function.

After 3 hours at a shopping mall, the average amount spent is $160.

( ) 42.2(1.56)xf x

( ) 42.2(1.56)xf x 3(3) 42.2(1.56)f 160.20876 160


Example: Graphing an Exponential Function

Graph: One of the ways to graph it:

“T” chart - plot points

( ) 3xf x

x

–2

–1

0

1

( ) 3xf x

2 1( 2) 3

9f

1 1( 1) 3

3f

0(0) 3 1f

1(1) 3 3f

We set up a table of coordinates, then plot these points, connecting them with a smooth, continuous curve.

-4 -3 -2 -1 1 2 3 4

-5

-4

-3

-2

-1

1

2

3

4

5

x

y


Characteristics of Exponential Functions of the Form ( ) xf x b


Example: Transformations Involving Exponential Functions

Use the graph of to obtain the graph of

Begin with

We’ve identified three

points and the asymptote.

( ) 3xf x 1( ) 3xg x

-4 -3 -2 -1 1 2 3 4

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

Horizontalasymptotey = 0

11,

3

(0,1)

(1,3)( ) 3xf x


Example: Transformations Involving Exponential Functions (continued)

Use the graph of to obtain the graph of

The graph will shift

1 unit to the right.

Add 1 to each

x-coordinate.

( ) 3xf x 1( ) 3xg x

-4 -3 -2 -1 1 2 3 4

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

10,

3

(2,3)

(1,1)

Horizontalasymptotey = 0

(0,1)

(1,3)

11,

3


The Natural Base e

The number e is defined as the value that

approaches as n gets larger and larger. As

the approximate value of e to nine decimal places is

The irrational number, e, approximately 2.72, is called the natural base. The function is called the natural exponential function.

11

n

n

n

2.718281827e

( ) xf x e


Example: Evaluating Functions with Base e

The exponential function models the gray wolf population of the Western Great Lakes, f(x), x years after 1978. Project the gray wolf’s population in the recovery area in 2012.

Because 2012 is 34 years after 1978, we substitute 34 for x in the given function.

This indicates that the gray wolf population in the Western Great Lakes in the year 2012 is projected to

be approximately 4446.

0.042( ) 1066 xf x e

0.042( ) 1066 xf x e 0.042(34)(34) 1066 4446f e


Formulas for Compound Interest

After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas:

1. For n compounding periods per year:

2. For continuous compounding:

1nt

rA P

n

rtA Pe


Example: Using Compound Interest Formulas

A sum of $10,000 is invested at an annual rate of 8%. Find the balance in the account after 5 years subject to quarterly compounding.

We will use the formula for n compounding periods per year, with n = 4.

The balance of the account after 5 years subject to quarterly compounding will be $14,859.47.

1nt

rA P

n

4 50.08

10,000 14

A

14,859.47


Example: Using Compound Interest Formulas

A sum of $10,000 is invested at an annual rate of 8%. Find the balance in the account after 5 years subject to continuous compounding.

We will use the formula for continuous compounding.

The balance in the account after 5 years subject to continuous compounding will be $14,918.25.

rtA Pe 0.08(5)10,000A e 14,918.25

Documents

Chapter 3 Exponential and Logarithmic Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 3.1 Exponential Functions