# Exponential Functions - PowerPoint PPT Presentation

Title:

## Exponential Functions

Description:

### Exponential Functions – PowerPoint PPT presentation

Number of Views:162
Avg rating:3.0/5.0
Slides: 15
Provided by: 00154
Category:
Tags:
Transcript and Presenter's Notes

Title: Exponential Functions

1
Exponential Functions
2
• A function that can be expressed in the form
• and is positive,
is called an Exponential Function.

3
Vocabulary Asymptote
• Exponential functions have a graph characteristic
called an asymptote.
• An asymptote is an imaginary line the graphs
infinite behavior approaches.

4
• 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.

5
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.

6
• 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?
7
• How would you graph

Is this graph increasing or decreasing?
Decreasing.
• Notice that the reflection is decreasing, so the
end behavior is

8
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

9
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
10
More Transformations
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
11
More Transformations
• In general, transformations of the exponential
parent function
• involve some combination of the following
formula

12
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

13
Domain Range Y-intercept H.A.
Continuous Increasing No vertical asymptotes
and
14
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