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6.2

6.2. REFLECTIONS AND SYMMETRY. 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. Vertical Symmetry.

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6.2

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  1. 6.2 REFLECTIONS AND SYMMETRY

  2. 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.

  3. 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 Algebraically, this means g(x)= − f (x). y = f(x) x y = g(x)

  4. 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 64 Algebraically, this means h(x) = f (− x). y = h(x) y = f(x) 32 x 16

  5. 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 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

  6. 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.

  7. Example of an Even Function Graph of y = f(x) = x2 – 4 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.

  8. 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.

  9. Example of an Odd Function Graph of y = f(x) = x3 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.

  10. Symmetric to y-axis Symmetric to origin

  11. Combining Shifts and Reflections Graph of y = f(x) = 1/x and y = g(x) = 4 – 1/x 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

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