Function-Graphing-Rules

Function-Graphing-Rules - Function Graphing Rules Are you...

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From MathMotivation.com – Permission Granted For Use and Modification For Non-Profit Purposes Function Graphing Rules Are you tired of calculating points? The rules in this section will allow you to graph something like y = 2(x –4) 2 – 3 without calculating a single point! Also, you will learn rules that will give you insight into the symmetry of graphs like y = x 4 – 2x 2 without calculating a point! Even Functions A function is defined as even if opposite real values of x result in the same y-value. In other words, a function is even if f(a) = f(-a) for any real value of “a”. For example, f(x) = x 2 is an even function since f(a) = a 2 and f(-a) = a 2 . Graphs of Even Functions Since opposite values of x result in the same y-value, the graph of an even function will always have symmetry with respect to the y-axis . For example, the graph of y=x 2 is shown below. Example: Plot the graph of y = x 4 – x 2 . Then determine whether or not this is an even function and note any symmetry of the graph. Letting x=0 results in y=0, so our y-intercept is (0,0). We can find the x-intercepts by letting y=0 and solving 0 = x 4 – x 2 . Factor 0 = x 4 – x 2 with the Distributive Property to get 0 = x 2 (x 2 – 1) = x 2 (x + 1)(x - 1). Apply the Zero Product Law to get solutions x 2 = 0, x + 1 = 0, and x - 1 = 0. Extract square roots and apply the Addition Property of Equality to solve and get x = 0, x = -1, x = 1. This results in x-intercepts of (0,0), (-1,0), and (1,0). Plot these points and find more points on each side of each x-intercept. Draw a smooth curve through as shown on the following page.
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From MathMotivation.com – Permission Granted For Use and Modification For Non-Profit Purposes It becomes clear that this is an even function as x=2 and x=-2 both resulted in y=12 and both x=1/2 and x=-1/2 both resulted in y = -3/16. Also, both terms of y = x 4 – x 2 involve even powers of x, making opposite input values of x result in the same y- value . The graph of this function clearly exhibits symmetry with respect to the y-axis.
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This note was uploaded on 11/05/2011 for the course ANTHRO 101 taught by Professor Crandall during the Fall '09 term at BYU.

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Function-Graphing-Rules - Function Graphing Rules Are you...

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