Quadratics and inequalities

4 12 10 10 calculator close up dug22241ch10aqxd

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4 12 10 10 Calculator Close-Up dug22241_ch10a.qxd 11/10/2004 18:31 Page 643
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Applications In applications we are often interested in finding the maximum or minimum value of a variable. If the graph of a parabola opens downward, then the maximum value of the dependent variable is the second coordinate of the vertex. If the parabola opens upward, then the minimum value of the dependent variable is the second coordinate of the vertex. 644 Chapter 10 Quadratic Equations and Inequalities 10-28 E X A M P L E 6 Finding the maximum height If a projectile is launched with an initial velocity of v 0 feet per second from an initial height of s 0 feet, then its height s ( t ) in feet is determined by s ( t ) 16 t 2 v 0 t s 0 , where t is the time in seconds. If a ball is tossed upward with velocity 64 feet per second from a height of 5 feet, then what is the maximum height reached by the ball? Solution The height s ( t ) of the ball for any time t is given by s ( t ) 16 t 2 64 t 5. Because the maximum height occurs at the vertex of the parabola, we use t 2 a b to find the vertex: t 2( 6 1 4 6) 2 Now use t 2 to find the second coordinate of the vertex: s (2) 16(2) 2 64(2) 5 69 The maximum height reached by the ball is 69 feet. See Fig. 10.8. Now do Exercises 59–67 Figure 10.8 Warm-Ups True or false? Explain your answer. 1. The ordered pair ( 2, 1) satisfies f ( x ) x 2 5. 2. The y -intercept for g ( x ) x 2 3 x 9 is (9, 0). 3. The x -intercepts for y x 2 5 are ( 5, 0 ) and ( 5, 0 ) . 4. The graph of f ( x ) x 2 12 opens upward. 5. The graph of y 4 x 2 opens downward. 6. The vertex of y x 2 2 x is ( 1, 1). 7. The parabola y x 2 1 has no x -intercepts. 8. The y -intercept for g ( x ) ax 2 bx c is (0, c ). 9. If w 2 v 2 9, then the maximum value of w is 9. 10. If y 3 x 2 7 x 9, then the maximum value of y occurs when x 7 6 . dug22241_ch10a.qxd 11/10/2004 18:31 Page 644
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10-29 10.3 Graphing Parabolas 645 Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What equation has a graph called a parabola? 2. When does a parabola open upward and when does a parabola open downward? 3. How can you find the x -intercepts for a parabola? 4. How can you find the y -intercept? 5. What is the x -coordinate of the vertex of a parabola? 6. How do you find the y -coordinate of the vertex? Complete each ordered pair so that it satisfies the given equation. See Example 1. 7. y x 2 x 12 (3, ), ( , 0) 8. y 1 2 x 2 x 1 (0, ), ( , 3) 9. y 16 x 2 32 x (4, ), ( , 0) 10. y x 2 4 x 5 ( 2, ), ( , 2) Determine whether the graph of each parabola opens upward or downward. See Examples 2 and 3. 11. y x 2 5 12. y 2 x 2 x 1 13. y 3 x 2 4 x 2 14. y x 2 3 15. y ( 2 x 3) 2 16. y (5 x ) 2 Graph each parabola. See Examples 2 and 3. 17. y x 2 2 18. y x 2 4 19. y 1 2 x 2 4 20. y 1 3 x 2 6 21. y 2 x 2 5 10.3 Exercises Boost your GRADE at mathzone.com! Practice Problems Self-Tests Videos Net Tutor e-Professors dug22241_ch10a.qxd 11/10/2004 18:31 Page 645
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646 Chapter 10 Quadratic Equations and Inequalities 10-30 22. y x 2 1 23. y 1 3 x 2 5 24. y 1 2 x 2 3 25. y ( x 2) 2 26. y ( x 3) 2 Find the vertex for the graph of each parabola. See Example 4.
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