Exponential Functions

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

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Vocabulary Asymptote

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

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• 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.

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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?
7

• 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 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

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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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More Transformations

• In general, transformations of the exponential
  parent function
• involve some combination of the following
  formula

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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

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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