# Exponential Functions - PowerPoint PPT Presentation

PPT – Exponential Functions PowerPoint presentation | free to download - id: 165907-ZDc1Z

The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
Title:

## Exponential Functions

Description:

### Exponential Functions with positive values of x are increasing, one-to-one functions. ... The letter e is the initial of the last name of Leonhard Euler (1701-1783) ... – PowerPoint PPT presentation

Number of Views:949
Avg rating:3.0/5.0
Slides: 16
Provided by: lw64
Category:
Transcript and Presenter's Notes

Title: Exponential Functions

1
Exponential Functions
L. Waihman 2002
2
• A function that can be expressed in the form
• and is positive,
is called an Exponential Function.
• Exponential Functions with positive values of x
are increasing, one-to-one functions.
• The parent form of the graph has a y-intercept at
(0,1) and passes through (1,b).
• The value of b determines the steepness of the
curve.
• The function is neither even nor odd. There is
no symmetry.
• There is no local extrema.

3
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.

4
• 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?
5
• Recall that if then the graph
of is a reflection of about the
y-axis.
• Thus, if then

Domain Range Y-intercept
Horizontal Asymptote
Concavity?
up
6
• How would you graph

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

7
How does b affect the function?
• If bgt1, then
• f is an increasing function,
• and
• If 0ltblt1, then
• f is a decreasing function,
• and

8
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
9
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
10
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
Domain Range Y-intercept H.A.
Continuous Increasing No vertical asymptotes
and
12
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
13
Exponential Equations
• Equations that contain one or more exponential
expressions are called exponential equations.
• Steps to solving some exponential equations
• Express both sides in terms of same base.
• When bases are the same, exponents are equal.
• i.e.

14
Exponential Equations
• Sometimes it may be helpful to factor the
equation to solve

or
There is no value of x for which is equal
to 0.
15
Exponential Equations
• Try
• 1) 2)

or