Symmetry with respect to x-axis
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Transforming Graphs,
Exponential and Logarithm
Functions
A graph is said to be Symmetric with respect to
the x-axis if, for every point (x, y) on the graph,
the point (x, -y) is also on the graph.
Peter Lo
M014 © Peter Lo 2002
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M014 © Peter Lo 2002
Symmetry with respect to y-axis
Symmetry with respect to origin
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A graph is said to be Symmetric with respect to
the y-axis if, for every point (x, y) on the graph,
the point (-x, y) is also on the graph.
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A graph is said to be Symmetric with respect to
the origin if, for every point (x, y) on the graph,
the point (-x, -y) is also on the graph.
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Reflection
Translation
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The graph of y = –f (x) is a reflection in the x-axis
of the graph of y = f (x).
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Translating Upward or Downward
u If k > 0, then the graph of y = f(x) + k is an Upward
Translation of the graph of y = f(x) and the graph of y
= f(x) – k is a Downward Translation of the graph of
y = f(x)
Translating to the Right and Left
u If h > 0, then the graph y = f(x – h) is a Translation to
the Right of the graph of y = f(x), and the graph of y =
f(x + h) is a Translation to the Left of the graph of y =
f(x)
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M014 © Peter Lo 2002
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Stretching and Shrinking
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If a > 1, then the graph of y = af(x) is obtained by
Stretching the graph of y = f(x).
If 0 < a < 1, then the graph of y = af(x) is obtained
by Shrinking the graph of y = f(x).
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Example
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Example
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Example
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Exponential Functions
Graphing Exponential Function
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An Exponential Function is a function of the form
u f (x) = a x
where a > 0 and a ≠ 1
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One-to-One Property of
Exponential Functions
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Example
For a > 0 and a ≠ 1,
u If a m = an , then m = n
M014 © Peter Lo 2002
The graph of the function f (x) = ax
u Pass through the point (0, 1).
u Has domain (–∞, ∞) and range (0, ∞).
u Approaches, but does not touch, the x-axis.
u Increase from Left to Right if a > 1.
u Decrease from Left to Right if 0 < a < 1.
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Sketch the graph of f (x) = 2x-1 – 2
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Example
Logarithm Functions
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For a > 0 and a ≠ 1,
u y = loga x if and only if ay = x.
M014 © Peter Lo 2002
Graphing Logarithm Function
Example
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The graph of the function f (x) = loga x
u Pass through the point (1, 0).
u Has domain (0, ∞) and range (–∞, ∞).
u Approaches, but does not touch, the y-axis.
u Increase from Left to Right if a > 1.
u Decrease from Left to Right if 0 < a < 1.
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Sketch the graph of g(x) = log2 (x) and compare it to the
graph y = 2x.
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Example
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Properties of Logarithms
Evaluate log5 (625)
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Summary of Properties
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Changing the Base
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Example:
u Evaluate log7 99 using base-change formula
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Strategy for Solving Equations
Example
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Example
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References
Solve 42x = 33x-1 .
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Solve 3log8 (2x – 1) = 4.
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Algebra for College Students (Ch. 10 – 11)
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