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Exponential Functions

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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

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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
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
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
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