Exponential FunctionsExponential Functions

L. Waihman2002

A function that can be expressed in the A function that can be expressed in the form form

and is and is positivepositive, is called , is called an Exponential Function.an Exponential Function.

Exponential Functions with positive values Exponential Functions with positive values of x are increasing, one-to-one functions.of x are increasing, one-to-one functions.

The parent form of the graph has a y-The parent form of the graph has a y-intercept at (0,1) and passes through (1,b).intercept at (0,1) and passes through (1,b).

The value of b determines the steepness of The value of b determines the steepness of the curve.the curve.

The function is neither even nor odd. The function is neither even nor odd. There is no symmetry.There is no symmetry.

There is no local extrema.There is no local extrema.

( ) , 0, 1xf x a b a b b

( ) xf x a b

( ) xf x a b

The domain is The domain is The range isThe range is End Behavior: End Behavior:

AsAs AsAs

The y-intercept is The y-intercept is The horizontal asymptote isThe horizontal asymptote is

, . 0, .

, ( ) 0.x f x , ( ) .x f x

0.y

More Characteristics of

0,1 .

There is no x-intercept.There is no x-intercept. There are no vertical asymptotes.There are no vertical asymptotes. This is a continuous function.This is a continuous function. It is concave up.It is concave up.

How would you graph How would you graph ( ) 3 ?xf x

( ) 6 ?xf x

Domain:

Range:

Y-intercept:

Domain:

Range:

Y-intercept:

, 0,

0,1

, 0,

0,1

Inc/dec?

Inc/dec?

HorizontalAsymptote: 0y

HorizontalAsymptote: 0y

Concavity?

Concavity?

How would you graphHow would you graphup

increasing

increasingup

Recall that if then the graph of Recall that if then the graph of is a reflection of about the y-axis. is a reflection of about the y-axis.

Thus, if then Thus, if then

1( )

x

f xb

( ) ( ),g x f x ( )g x( )f x

( ) 2 ,xf x 1

( ) .2

x

g x

Domain:

Range:

Y-intercept:

, 0,

0,1HorizontalAsymptote: 0y Concavity? up

Notice that the reflection is Notice that the reflection is decreasing, so the end behavior is:decreasing, so the end behavior is:

, ( ) 0.x f x , ( ) .x f x

1( ) ?

3

x

f x

Is this graph increasingor decreasing?

Decreasing.

How would you graphHow would you graph

How does b affect the function?How does b affect the function?

If b>1, then•f is an increasing function,• andlim ( ) 0

xf x

lim ( )

xf x

If 0<b<1, then•f is a decreasing function,• andlim ( )

xf x

lim ( ) 0

xf x

TransformationsTransformations

• Exponential graphs, like other functions we have studied, can be dilated, reflected and translated.• It is important to maintain the same base as you analyze the transformations.

( ) 2 3xg x Vertical shift up 3

( ) 3(2 ) 1xg x Reflect @ x-axisVertical stretch 3Vertical shift down 1

More TransformationsMore Transformations1(2) 1xy

Reflect about the x-axis.Horizontal shift right 1.Vertical shift up 1.

212 (3) 3xy

Vertical shrink ½ .Horizontal shift left 2.

Vertical shift down 3.

Domain

: Range

:

Y-

intercept:

HorizontalAsymptote:Inc/dec?Concavity?

Domai

n:Range:

Y-intercept:

HorizontalAsymptote:

Inc/dec?Concavity?

, .

,1 .

120, .

1.y

decreasingdown

, .

3, .

3.y 3

20, .

increasingup

The number e

•The letter e is the initial of the last name of Leonhard Euler (1701-1783) who introduced the notation.

• Since has special calculus properties that simplify many calculations, it is the natural base of exponential functions.

•The value of e is defined as the number that the expression approaches as n approaches infinity.

• The value of e to 16 decimal places is 2.7182818284590452.

• The function is called the Natural Exponential Function

11

n

n

( ) xf x e

( ) xf x e

( ) xf x e

Domain:

Range:

Y-

intercept:

H.A.:

,

0,

0,1

0y

ContinuousIncreasingNo vertical asymptotes

lim 0x

xe

lim x

xe

and

( ) 3 2xf x e 2 2xf x e 1xf x e

Domain:

Range:

Y-intercept:

H.A.:

Domain:

Range:

Y-intercept:

H.A.:

Domain:

Range:

Y-intercept:

H.A.:

, , ,

2, 0,5

2y

, 1 0, 2

1y

2,

0,9.389

2y

TransformationsTransformations

Vertical stretch 3.Vertical shift up 2.

Reflect @ x-axis.Vertical shift down 1.

Horizontal shift left 2.Vertical shift up 2

Inc/dec? increasing

Concavity?up

Inc/dec? Inc/dec?decreasing

Concavity?down

increasing

Concavity? up

Exponential EquationsExponential Equations Equations that contain one or more Equations that contain one or more

exponential expressions are called exponential expressions are called exponential equationsexponential equations..

Steps to solving some exponential Steps to solving some exponential equations:equations:

1.1. Express both sides in terms of same base.Express both sides in terms of same base.2.2. When bases are the same, exponents are equal.When bases are the same, exponents are equal. i.e.: i.e.:

12 2

12 2

1

2

1

4

5 5

5 5

2

x

x

x

x

Exponential EquationsExponential Equations

Sometimes it may be helpful to Sometimes it may be helpful to factor the equation to solve:factor the equation to solve:

2 4 0xx e

2 4 0x 0xe 2 4x

2x There is no value of x forwhich is equal to 0.

xe

or

2 4x xxe e

2 4 0x xxe e

Exponential EquationsExponential Equations

Try:Try: 1) 2) 1) 2) 3 14 32x 2 3 34x xx e e

3 12 52 2x

6 2 52 2x

6 2 5x

6 3x 1

2x

2 3 34 0x xx e e

3 2 4 0xe x 3 0xe or

2 4 0x 2 4x

2x