determine whether a graph is symmetric with respect to the x-axis, the y-axis, and the origin....
TRANSCRIPT
Determine whether a graph is symmetric with respect to the x-axis, the y-axis, and the origin.
Determine whether a function is even, odd, or neither even nor odd.
Given the graph of a function, graph its transformation using horizontal and vertical shift, reflection, stretching, and shrinking.
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2.4 Symmetry and Transformations
Symmetry
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Algebraic Tests of Symmetryx-axis: If replacing y with y produces an equivalent equation, then the graph is symmetric with respect to the x-axis.y-axis: If replacing x with x produces an equivalent equation, then the graph is symmetric with respect to the y-axis.Origin: If replacing x with x and y with y produces an equivalent equation, then the graph is symmetric with respect to the origin.
Example
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Test y = x2 + 2 for symmetry with respect to the x-axis, the y-axis, and the origin.
x-axis: We replace y with y y-axis: We replace x with x
Origin: We replace x with x and y with y:
Example 2
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Using the Graph to Find Symmetry
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Even and Odd Functions
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If the graph of a function f is symmetric with respect to the y-axis, we say that it is an even function. That is, for each x in the domain of f, f(x) = f(x).
If the graph of a function f is symmetric with respect to the origin, we say that it is an odd function. That is, for each x in the domain of f, f(x) = f(x).
Example
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Determine whether the function is even, odd, or neither.
4 2( ) 4h x x x
y = x4 4x2
Example 4
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Basic Graph of Functions
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Vertical Translation
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Vertical Translation
For b > 0, the graph of
y = f(x) + b
is the graph of y = f(x) shifted up b units.
The graph of
y = f(x) b
is the graph of y = f(x) shifted down b units.
y = 3x2 – 3
y = 3x2
y = 3x2 +2
Horizontal Translation
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Horizontal Translation
For d > 0, the graph of
y = f(x d)
is the graph of y = f(x) shifted right d units;
The graph of
y = f(x + d)
is the graph of y = f(x) shifted left d units.
y = 3x2
y = (3x – 2)2
y = (3x + 2)2
Examples
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Reflections
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The graph of y = f(x) is the reflection of the graph of y = f(x) across the x-axis.
The graph of y = f(x) is the reflection of the graph of y = f(x) across the y-axis.
Example Graphs
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Example
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Reflection of the graph y = x3 2x2 across the y-axis.
y = x3 2x2y = -x3 + 2x2
Vertical Stretching and Shrinking
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The graph of y = af (x) can be obtained from the graph of y = f(x) by
stretching vertically for |a| > 1, orshrinking vertically for 0 < |a| < 1.
For a < 0, the graph is also reflected across the x-axis.
(The y-coordinates of the graph of y = af (x) can be obtained by multiplying the y-coordinates of y = f(x) by a.)
Examples
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Stretch y = x3 – x vertically.
Describe it with an Equation
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