Exponential Functions

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Exponential Functions

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• The Math Center

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Exponential Functions

• Def An exponential function with base a is a
  function of the form f(x) ax , where a and x
  are real numbers such that a gt 0 and a ? 1.
• i.e. The base is constant and the exponent is a
  variable.

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Properties of Exponential Functions

1. f(x) is increasing for a gt 1 and decreasing for
   0 lt a lt 1.
2. The y-intercept is at (0,1).
3. The x-axis is a horizontal asymptote.
4. Has domain (-8, 8) and range (0, 8).
5. f(x) is a one-to-one function.

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• Exponential Family of Functions (Graphing)

• y abxc d
• By Table(pick a number according to properties
  listed above, use a calculator if necessary)
• (2) By Transformation (translating, stretching,
  shrinking and reflecting)

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Graphing

• Example the graph of y 2x
• Notice the y-intercept is at (0,1)
• The x-axis is a horizontal Asymptote, i.e. the
  line which the graph of y 2x approaches as x? -
  
• but never crosses.

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Graphing

• Graph of

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Graphing

• Translate (i.e. shifted) 3 units to the left
• The domain is
• (-8,8).

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Logarithmic Functions

• Def For a gt 0 and a ? 1 , the logarithmic
  function with base a is denoted
• f(x) loga(x), where y loga(x) if and only if
  ay x .
• i.e. instead of f -1(x) we use loga(x) as the
  inverse of an exponential function

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Two Popular Bases

• Common logbase 10 (log)
• Natural logbase e (ln)
• Properties of Logarithmic Functions
• f(x) is increasing for a gt1 and decreasing for
  0ltalt 1
• The x-intercept is at (1,0)
• The y-axis is a vertical asymptote
• Has Domain (0,8) and Range (-8,8)
• f(x) is a one-to-one function

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Logarithmic Family of Functions (Graphing)
Graphing Logarithms

1. By Setting up table
2. By Transformation

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Example
y log (x)

• The common Log is base 10, when the base a is not
  written, it is understood to be base 10.
• The x-intercept (i.e. zero or root) is (1,0)
• The vertical asymptote is y-axis

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Graph of Logarithmic Functions
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Exponential Functions vs Logarithmic Functions

• The exponential function ex and Natural Log (Ln)
  are inverses of each other, notice the reflection
  over the diagonal line y x
• Note e 2.71828
• Note Ln rises more rapidly than the common log

v.s
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Example
y log (x 1)

• Shifted one points to the right, (x-1)
• New vertical asymptote at x 1
• Zero at (2, 0)
• Domain is from (-8,8)

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Graph of a Logarithmic Transformation
y log (x - 1)
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Exponential and Logarithmic Functions Links

• Exponential Functions Handout
• Algebra and Logarithmic Functions Handout
• Exponential Functions Quiz