Exponential Functions

- A function that can be expressed in the form
- and is positive,

is called an Exponential Function.

Vocabulary Asymptote

- Exponential functions have a graph characteristic

called an asymptote. - An asymptote is an imaginary line the graphs

infinite behavior approaches.

- Exponential Function
- Graph Characteristics
- The parent exponential function has a y-intercept

at (0,1). - The parent exponential function has an asymptote

at y 0. - The value of b determines the steepness of the

curve. - There are no local extrema.

More Characteristics of

- The domain is
- The range is
- End Behavior
- As
- As
- The y-intercept is
- The horizontal asymptote is

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

- How would you graph

Domain Range Y-intercept

Horizontal Asymptote

Inc/dec?

increasing

Concavity?

up

- How would you graph

Domain Range Y-intercept

Horizontal Asymptote

Inc/dec?

increasing

up

Concavity?

- How would you graph

Is this graph increasing or decreasing?

Decreasing.

- Notice that the reflection is decreasing, so the

end behavior is

How do a and b affect the function?

- 4 typse of graphs
- a and bgt1, then f is an exponential growth
- -a and b gt1, then f is reflected down

- 3) a and 0ltblt1, then f is an exponential decay
- 4) a and 0ltblt1, then f is reflected down

Transformations

- 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.

Reflect _at_ x-axis Vertical stretch 3 Vertical

shift down 1

Vertical shift up 3

More Transformations

Reflect about the x-axis.

Vertical shrink ½ .

Horizontal shift left 2.

Horizontal shift right 1.

Vertical shift up 1.

Vertical shift down 3.

Domain

Domain

Range

Range

Horizontal Asymptote

Horizontal Asymptote

Y-intercept

Y-intercept

Inc/dec?

Inc/dec?

decreasing

increasing

Concavity?

Concavity?

down

up

More Transformations

- In general, transformations of the exponential

parent function - involve some combination of the following

formula

The number e

- The number e has the value 2.71828 and possesses

special calculus properties that simplify many

calculations and is also called the natural base

of exponential functions. - The function is called the Natural

Exponential Function

Domain Range Y-intercept H.A.

Continuous Increasing No vertical asymptotes

and

Transformations

Vertical stretch 3.

Horizontal shift left 2.

Reflect _at_ x-axis.

Vertical shift up 2

Vertical shift up 2.

Vertical shift down 1.

Domain Range Y-intercept H.A.

Domain Range Y-intercept H.A.

Domain Range Y-intercept H.A.

Inc/dec?

increasing

Inc/dec?

increasing

Inc/dec?

decreasing

Concavity?

up

Concavity?

up

Concavity?

down