Exponential Exponential FunctionsFunctions

Exponential Functions and Their Graphs

Irrational Irrational ExponentsExponents

If b is a positive number and x is a real number, the expression bx always represents a positive number. It is also true that the familiar properties of exponents hold for irrational exponents.

Example 1:Example 1:

Use properties of exponents to Use properties of exponents to simplify simplify

22) 3a

Example 1:Example 1:

Use properties of exponents to simplify Use properties of exponents to simplify

22 2 2

4

2

) 3 3

3

3

9

a

Example 1:Example 1:

Use properties of exponents to simplify Use properties of exponents to simplify

8 2)b a a

Example 1:Example 1:

Use properties of exponents to simplify Use properties of exponents to simplify

8 2 8 2

4 2 2

2 2 2

3 2

)b a a a

a

a

a

Exponential FunctionsExponential Functions

An exponential function with base An exponential function with base b is defined by the equationb is defined by the equation

x is a real number.x is a real number. The domain of any exponential The domain of any exponential

function is the intervalfunction is the interval

The range is the interval The range is the interval

0 , 1xf x b where b b and

,

0,

Graphing Exponential Graphing Exponential FunctionsFunctions

2xGraph f x

Graphing Exponential Graphing Exponential FunctionsFunctions

1

2

x

Graph f x

Example 2:Example 2:

4xGraph f x

Let’s make a table and plot points to graph.

Example 2:Example 2:

4xGraph f x

Example 2:Example 2:

4xGraph f x

Properties:Properties:

Exponential FunctionsExponential Functions

Example 3:Example 3:

Given a graph, find the value of b:Given a graph, find the value of b:

Example 3:Example 3:

Given a graph, find the value of b:Given a graph, find the value of b:

Increasing and Increasing and Decreasing Decreasing FunctionsFunctions

One-to-One One-to-One Exponential FunctionsExponential Functions

Compound InterestCompound Interest

1kt

rA P

k

Example 4:Example 4:

The parents of a newborn child invest $8,000 in a plan that earns 9% interest, compounded quarterly. If the money is left untouched, how much will the child have in the account in 55 years?

Example 4 Solution:Example 4 Solution:

4 55

220

1

0.098000 1

4

8000 1.0225

$ 1,069,103.27

ktr

A Pk

A

Using the compound interest formula:

Future value of account in 55 years

Base Base e e Exponential FunctionsExponential Functions

Sometimes called the natural base, Sometimes called the natural base,

often appears as the base of an often appears as the base of an exponential functions. exponential functions.

It is the base of the continuous It is the base of the continuous compound interest formula: compound interest formula:

2.71828182845....e irrational number

rtA Pe

Example 5:Example 5:

If the parents of the newborn child in Example 4 had invested $8,000 at an annual rate of 9%, compounded continuously, how much would the child have in the account in 55 years?

Example 5 Example 5 Solution:Solution:

0.09 55

4.95

8000

8000

$ 1,129,399.71

rtA Pe compounded continuously

A e

e

Future value of account in 55 years

Graphing Graphing

Make a table and plot points:Make a table and plot points:

xf x e

Exponential Exponential FunctionsFunctions

Horizontal asymptoteHorizontal asymptote Function increasesFunction increases y-intercept (0,1)y-intercept (0,1) Domain all real Domain all real

numbersnumbers Range: y > 0Range: y > 0

TranslationsTranslations

For k>0 For k>0 y = f(x) + k y = f(x) + k y = f(x) – ky = f(x) – k y = f(x - k)y = f(x - k) y = f(x + k)y = f(x + k)

Up k unitsDown k unitsRight k unitsLeft k units

Example 6:Example 6:

On one set of axes, graph On one set of axes, graph

2 2 3x xf x and f x

Example 6:Example 6: On one set of axes, graph On one set of axes, graph

2 2 3x xf x and f x

Up 3

Example 7:Example 7: On one set of axes, graph On one set of axes, graph

3x xf x e and f x e

Right 3

Non-Rigid Non-Rigid TransformationsTransformations Exponential Functions with the Exponential Functions with the

form f(x)=kbform f(x)=kbxx and f(x)=b and f(x)=bkx kx are vertical and horizontal stretchings of the graph f(x)=bx. Use a graphing calculator to graph these functions.