28 CHAPTER 1 Graphs Applications and Extensions 81. Given that the point ( ) 1, 2 is on the graph of an equation that is symmetric with respect to the origin, what other point is on the graph? 82. If the graph of an equation is symmetric with respect to the y -axis and 6 is an x -intercept of this graph, name another x -intercept. 83. If the graph of an equation is symmetric with respect to the origin and −4 is an x -intercept of this graph, name another x -intercept. 84. If the graph of an equation is symmetric with respect to the x -axis and 2 is a y -intercept, name another y -intercept. 85. Microphones In studios and on stages, cardioid microphones are often preferred for the richness they add to voices and for their ability to reduce the level of sound from the sides and rear of the microphone. Suppose one such cardioid pattern is given by the equation ( ) + − = + x y x x y . 2 2 2 2 2 (a) Find the intercepts of the graph of the equation. (b) Test for symmetry with respect to the x -axis, the y -axis, and the origin. Source: www.notaviva.com 86. Solar Energy The solar electric generating systems at Kramer Junction, California, use parabolic troughs to heat a heat-transfer fluid to a high temperature. This fluid is used to generate steam that drives a power conversion system to produce electricity. For troughs 7.5 feet wide, an equation for the cross section is = − y x 16 120 225. 2 (a) Find the intercepts of the graph of the equation. (b) Test for symmetry with respect to the x -axis, the y -axis, and the origin. Source: U.S. Department of Energy 87. Challenge Problem Lemniscate For a nonzero constant a, find the intercepts of the graph of ( ) ( ) + = − x y a x y . 2 2 2 2 2 2 Then test for symmetry with respect to the x -axis, the y -axis, and the origin. 88. Challenge Problem Limaçon For nonzero constants a and b, find the intercepts of the graph of x y ax b x y 2 2 2 2 2 2 ( ) ( ) + − = + Then test for symmetry with respect to the x -axis, the y -axis, and the origin. Explaining Concepts 89. Draw a graph of an equation that contains two x -intercepts; at one the graph crosses the x -axis, and at the other the graph touches the x -axis. 90. An equation is being tested for symmetry with respect to the x -axis, the y -axis, and the origin. Explain why, if two of these symmetries are present, the remaining one must also be present. 91. Draw a graph that contains the points ( ) ( ) − − 2,5 , 1,3 , and ( ) 0, 2 and is symmetric with respect to the y -axis. Compare your graph with those of other students; comment on any similarities. Can a graph contain these points and be symmetric with respect to the x -axis? the origin? Why or why not? ‘Are You Prepared?’ Answers 1. { } −6 2. { } −2, 6 1.4 Solving Equations Using a Graphing Utility Now Work the ‘Are You Prepared?’ problems on page 31. • Solve Linear, Quadratic, and Rational Equations (Section A.6, pp. A45–A53) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVE 1 Solve Equations Using a Graphing Utility (p. 29) One goal of this text is to determine when equations can be solved algebraically. If an algebraic method for solving an equation exists, we shall use it to obtain exact solutions.A graphing utility can then be used to support the algebraic result. However, if no algebraic techniques are available to solve an equation, a graphing utility will be used to obtain approximate solutions. Credit: stockyimages/Shutterstock Credit: U.S. Department of Energy
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