26 CHAPTER 1 Graphs if x is a large and positive number, then = y x 1 is a positive number close to 0. We also infer that if x is a positive number close to 0, then = y x 1 is a large and positive number. Armed with this information, we can graph the equation. Figure 41(a) illustrates some of these points and the graph of = y x 1 . Observe how the absence of intercepts and the existence of symmetry with respect to the origin were utilized. Figure 41(b) confirms our algebraic analysis using a TI-84 Plus CE. (a) y x 1 = x y 3 3 –3 –3 , 2 – 1 –– 2 (1, 1) 2, 1 –– 2 –2, ( ) ( ) ( ) ( ) 1 –– 2 (–1, –1) , –2 1 –– 2 – (b) y x 1 = 4 24 23 3 Y1 5 x 1 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 1.3 Assess Your Understanding 1. Solve: ( ) + − = − x 2 3 1 7 (pp. A45–A47) 2. Solve: − − = x x4 12 0 2 (pp. A47–A53) 3. The points, if any, at which a graph crosses or touches the coordinate axes are called . 4. The x -intercepts of the graph of an equation are those x -values for which . 5. If for every point ( ) x y , on the graph of an equation the point ( ) −x y , is also on the graph, then the graph is symmetric with respect to the . 6. If the graph of an equation is symmetric with respect to the y -axis and −4 is an x -intercept of this graph, then is also an x -intercept. 7. If the graph of an equation is symmetric with respect to the origin and ( ) − 3, 4 is a point on the graph, then is also a point on the graph. 8. True or False To find the y -intercepts of the graph of an equation, let = x 0 and solve for y. 9. True or False The y -coordinate of a point at which the graph crosses or touches the x -axis is an x -intercept. 10. True or False If a graph is symmetric with respect to the x -axis, then it cannot be symmetric with respect to the y -axis. 11. Multiple Choice Given that the intercepts of a graph are ( ) −4, 0 and ( ) 0, 5 , choose the statement that is true. (a) The y -intercept is −4, and the x -intercept is 5. (b) The y -intercepts are −4 and 5. (c) The x -intercepts are −4 and 5. (d) The x -intercept is −4, and the y -intercept is 5. 12. Multiple Choice To test whether the graph of an equation is symmetric with respect to the origin, replace in the equation and simplify. If an equivalent equation results, then the graph is symmetric with respect to the origin. (a) x by −x (b) y by −y (c) x by −x and y by −y (d) x by −y and y by −x Concepts and Vocabulary In Problems 13–24, find the intercepts and graph each equation by plotting points. Be sure to label the intercepts. 13. = + y x 2 14. = − y x 6 15. = + y x2 8 16. = − y x3 9 17. = − y x 1 2 18. = − y x 9 2 19. = − + y x 4 2 20. = − + y x 1 2 21. + = x y 2 3 6 22. + = x y 5 2 10 23. + = x y 9 4 36 2 24. + = x y 4 4 2 Skill Building 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure Figure 41
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