SECTION 1.3 Intercepts; Symmetry; Graphing Key Equations 23 Finding Intercepts and Testing an Equation for Symmetry For the equation = − + y x x 4 1 : 2 2 (a) find the intercepts and (b) test for symmetry. Solution (a) To obtain the x-intercept(s), let = y 0 in the equation and solve for x. − + = − = = − = x x x x x 4 1 0 4 0 2 or 2 2 2 2 To obtain the y-intercept(s), let = x 0 in the equation and solve for y. = − + = − + = − = − y x x 4 1 0 4 0 1 4 1 4 2 2 2 2 The x-intercepts are −2 and 2; the y-intercept is −4. (b) Now test the equation for symmetry with respect to the x-axis, the y-axis, and the origin. x-Axis: To test for symmetry with respect to the x-axis, replace y by −y. Since y x x y x x 4 1 is not equivalent to 4 1 2 2 2 2 − = − + = − + the graph of the equation is not symmetric with respect to the x-axis. y-Axis: To test for symmetry with respect to the y-axis, replace x by −x. Since − − ( ) ( ) = − + = − + = − + y x x x x y x x 4 1 4 1 is equivalent to 4 1 2 2 2 2 2 2 the graph of the equation is symmetric with respect to the y-axis. Origin: To test for symmetry with respect to the origin, replace x by −x and y by −y. − − − ( ) ( ) = − + − = − + y x x y x x 4 1 4 1 2 2 2 2 Replace x by x− and y by y. − Simplify. Since the result is not equivalent to the original equation, the graph of the equation = − + y x x 4 1 2 2 is not symmetric with respect to the origin. EXAMPLE 6 Let y 0. = Multiply both sides by x 1. 2 + Factor and use the Zero-Product Property. Seeing the Concept Figure 35 shows the graph of + = x y 4 9 36 2 2 using Desmos. Do you see the symmetry with respect to the x-axis, the y-axis, and the origin? Figure 35 x y 4 9 36 2 2 + = Seeing the Concept Figure 36 shows the graph of = − + y x x 4 1 2 2 using a TI-84 Plus CE graphing calculator. Do you see the symmetry with respect to the y-axis? Also, did you notice that the point ( ) 2, 0 is on the graph along with ( ) −2, 0 ? How could we have found the second x-intercept using symmetry? Figure 36 y x x 4 1 2 2 = − + 2 25 25 5 Now Work PROBLEM 55
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