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REFLECTIONS AND SYMMETRY
6.2 REFLECTIONS AND SYMMETRY
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A Formula for a Reflection
For a function f: • The graph of y = −f(x) is a reflection of the graph of y = f(x) about the x-axis. • The graph of y = f(−x) is a reflection of the graph of y = f(x) about the y-axis.
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Vertical Symmetry Example 1 (b) The graph shows a function f(x) in blue and its reflection through the x-axis in red (g(x)). The table shows values for f(x) and g(x). Write a formula for g(x) in terms of f(x) Solution (b) y x f (x) g(x) -3 1 -1 -2 2 4 -4 8 -8 16 -16 32 -32 3 64 -64 Algebraically, this means g(x) = − f (x). y = f(x) x y = g(x)
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Horizontal Symmetry Example 1 (b) The graph shows a function f(x) in blue and its reflection through the y-axis in red (h(x)). The table shows values for f(x) and h(x). Write a formula for h(x) in terms of f(x) Solution (b) y x f (x) h(x) -3 1 64 -2 2 32 -1 4 16 8 3 64 Algebraically, this means h(x) = f (− x). y = h(x) y = f(x) 32 x 16
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Horizontal and Vertical Reflection Combined
Example 1 (c) The graph shows a function f(x) in blue and its reflection through both the x-axis the y-axis in red (k(x)). The table shows values for f(x) and k(x). Write a formula for k(x) in terms of f(x) Solution (c) y x f (x) k(x) -3 1 -64 -2 2 -32 -1 4 -16 8 -8 16 -4 32 3 64 64 When a point is reflected about both the x-axis and the y-axis, both the x-value and the y-value change signs. Algebraically, this means k(x) = − f (− x). y = f(x) 32 x y = k(x) -32 -64
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Even Functions Are Symmetric About the y-Axis
If f is a function, then f is called an even function if, for all values of x in the domain of f, f(−x) = f(x). The graph of f is symmetric about the y-axis.
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Example of an Even Function
Consider the function f(x) = x2 – 4. Note that f(– x) = (– x)2 – 4 = x2 – 4 So f(x) = f(–x) . This means that f(x) is an even function. Graph of y = f(x) = x2 – 4
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Odd Functions Are Symmetric About the Origin
If f is a function, then f is called an odd function if, for all values of x in the domain of f, f(−x) = − f(x). The graph of f is symmetric about the origin.
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Example of an Odd Function
Consider the function f(x) = x3. Note that f(–x) = (– x)3 = – x3 So f(-x) = – f(x) . This means that f(x) is an odd function. Graph of y = f(x) = x3
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Symmetric to y-axis Symmetric to origin
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Combining Shifts and Reflections
Consider the function f(x) = 1/x. Consider the reflection of this function through the x-axis together with a shift of 4 units in the positive vertical direction g(x) = 4 – 1/x Graph of y = f(x) = 1/x and y = g(x) = 4 – 1/x
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