Warm-upGiven the function:

Find the inverse function and complete the table below.

x-interceptsy-interceptsvertical asymptoteshorizontal asymptotes

f 1f

Sketch the graph of both on the same set of axes

312)(

xxxf

4.3 Exponential Functions

Warm-upWhat are the characteristics of an exponential function, based on the following list of examples.

Examples of exponential functions.

NOT an Example of exponential function.

x2x10

xe x13 x

1

32

x 2x x1

x1

xxx

21

An exponential function is of the form where:

BASE is: a with requirement: a > 0 and a ≠ 1

EXPONENT is: x

I. Exponential Function

f (x) ax

ApplicationExample: The function:

describes the number of O-Rings expected to fail when the temperature is x (degrees F). The space-shuttle Challenger exploded in 1986 when the temperature was 31 o F .

What is the expected number of O-rings to fail at 31 o F ?

at 60 o F ?

1)967.0(49.13)( xxf

Do you recall these rules?

II. Laws of Exponents

yxaa

yxa

xab

xa 0a

Example: Plot points and sketch:

III. Graphing an Exponential Functionxxf 2)(

Example: Plot points and sketch: x

xf

21)(

III. Properties of the graph of Exponential Function

f (x) ax

Domain/Rangex-interceptsy-intercept

Horizontal Asymptotes

Increasing or Decreasing ?

Key Points (0,1), (1, 2) (0,1),(1, ½)

x2 :Examplex

Example

21 :

If base is greater than 1 If base is less than 1

IV. Transformations of the graph of Exponential Function

What do transformations do to the graph of an exponential function?

xy 2 xy 2 p. 283, # 29-36

IV. Transformations of the graph of Exponential Function

What do transformations do to the graph of

cay x

xay

cxay xay

xay xcay

cxay

V. Graphing on the Calculator

Graph

Does your graph demonstrate the correct shape for exponential growth?

If not, what happened?

xy 23

Watch your parentheses!!!!!

Enter y = 3^(2x) on the calculator

VI. A “Natural” Base. The Number e

First approximation for was found in studying continuous compound interest.

is the number given by the value of

n

n

n

as 11

e

e

The calculator gives us an approximation to the number.

Where does the graph of lie?

...718281828.21 e

xey

A Theorem for Exponential Functions

What does the base of an exponential function represent?

Rumors: Scenario 1: # people you tell is 2 per day.

Scenario 2: # people you tell is 3 per day.

What can you say about the ratio of consecutive days ?

Theorem: An exponential function satisfies the property:

xaxf )(

axf

xf

)(

)1(

A Theorem for Exponential Functions

How can we solve for x using the graphing calculator ?

Algebraically ?

800012 1 x

If an exponential equation can be expressed in the form: (same base on each side of equal sign)

Then we are HAPPY, ‘cause it’s pretty easy to solve

)(u(x) xvaa

If a problem is not written in this form, we can TRY to get it that way:

1422 1) xx

where u and v are expressions in x

If au av, then u v

xx 461 2)x

x3

812 3)

2

422 x

52 33 x

VII. Solving Exponential Equations

If a problem can be not written in this form, we will use logarithms to solve for the unknown

Exponential Equations with base eTreat as a number.

Solve for x: If we have a problem of the form:

3

2 1 2)2

eee xx

vuee v then ,u

e

124 2

1) eee xx

4ee x