3.1 Exponential Functions

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3.1 Exponential Functions
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Objective

• To graph exponential functions
• To solve simple exponential equations

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• 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.

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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.

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• 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?
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• 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
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• How would you graph

Is this graph increasing or decreasing?
Decreasing.

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

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How does b affect the function?

• If bgt1, then
• f is an increasing function,
• and

• If 0ltblt1, then
• f is a decreasing function,
• and

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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
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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
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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

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

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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.
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Exponential Equations

• Try
• 1) 2)

or