4_1 - Chapter 4 Polynomial and Rational Functions 4.1 1...

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Chapter 4 384 Polynomial and Rational Functions 4.1 Quadratic Functions and Models 1. x- intercepts: (–3,0), (3,0) y- intercept: (0,–9) 2. 2 x 2 + 7 x - 4 = 0 2 x - 1 ( 29 x + 4 ( 29 = 0 2 x - 1 = 0 x = 1 2 x + 4 = 0 x = - 4 The solution set is - 4, 1 2 . 3. - 5 2 2 = 25 4 4. right, 4 5. parabola 6. axis of symmetry 7. - b 2 a 8. True 9. False 10. True 11. C 12. E 13. F 14. A 15. G 16. B 17. H 18. D 19. f ( x ) = 1 4 x 2 Using the graph of y = x 2 , compress vertically by a factor of 1 4 .
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Section 4.1 Quadratic Functions and Models . 385 20. f ( x ) = 2 x 2 Using the graph of y = x 2 , stretch vertically by a factor of 2. 21. f ( x ) = 1 4 x 2 - 2 Using the graph of y = x 2 , compress vertically by a factor of 1 4 , then shift down 2 units. 22. f ( x ) = 2 x 2 - 3 Using the graph of y = x 2 , stretch vertically by a factor of 2, then shift down 3 units. 23. f ( x ) = 1 4 x 2 + 2 Using the graph of y = x 2 , compress vertically by a factor of 1 4 , then shift up 2 units.
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Chapter 4 Polynomial and Rational Functions 386 24. f ( x ) = 2 x 2 + 4 Using the graph of y = x 2 , stretch vertically by a factor of 2, then shift up 4 units. 25. f ( x ) = 1 4 x 2 + 1 Using the graph of y = x 2 , compress vertically by a factor of 1 4 , then shift up 1 unit. 26. f ( x ) = - 2 x 2 - 2 Using the graph of y = x 2 , stretch vertically by a factor of 2, reflect across the x- axis, then shift down 2 units. 27. f ( x ) = x 2 + 4 x + 2 = x 2 + 4 x + 4 ( 29 + 2 - 4 = x + 2 ( 29 2 - 2 Using the graph of y = x 2 , shift left 2 units, then shift down 2 units.
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Section 4.1 Quadratic Functions and Models . 387 28. f ( x ) = x 2 - 6 x - 1 = x 2 - 6 x + 9 ( 29 - 1 - 9 = x - 3 ( 29 2 - 10 Using the graph of y = x 2 , shift right 3 units, then shift down 10 units. 29. f ( x ) = 2 x 2 - 4 x + 1 = 2 x 2 - 2 x + 1 ( 29 + 1 - 2 = 2 x - 1 ( 29 2 - 1 Using the graph of y = x 2 , shift right 1 unit, stretch vertically by a factor of 2, then shift down 1 unit. 30. f ( x ) = 3 x 2 + 6 x = 3 x 2 + 2 x + 1 ( 29 - 3 = 3 x + 1 ( 29 2 - 3 Using the graph of y = x 2 , shift left 1 unit, stretch vertically by a factor of 3, then shift down 3 units. 31. f ( x ) = - x 2 - 2 x = - x 2 + 2 x + 1 ( 29 + 1 = - x + 1 ( 29 2 + 1 Using the graph of y = x 2 , shift left 1 unit, reflect across the x- axis, then shift up 1 unit.
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Chapter 4 Polynomial and Rational Functions 388 32. f ( x ) = - 2 x 2 + 6 x + 2 = - 2 x 2 - 3 x + 9 4 + 2 + 9 2 = - 2 x - 3 2 2 + 13 2 Using the graph of y = x 2 , shift right 3 2 units, reflect across the x- axis, stretch vertically by a factor of 2, then shift up 13 2 units. 33. f ( x ) = 1 2 x 2 + x - 1 = 1 2 x 2 + 2 x + 1 ( 29 - 1 - 1 2 = 1 2 x + 1 ( 29 2 - 3 2 Using the graph of y = x 2 , shift left 1 unit, compress vertically by a factor of 1 2 , then shift down 3 2 units. 34. f ( x ) = 2 3 x 2 + 4 3 x - 1 = 2 3 x 2 + 2 x + 1 ( 29 - 1 - 2 3 = 2 3 x + 1 ( 29 2 - 5 3 Using the graph of y = x 2 , shift left 1 unit, compress vertically by a factor of 2 3 , then shift down 5 3 units. 35. f ( x ) = x 2 + 2 x a = 1, b = 2, c = 0. Since a = 1 0, the graph opens up. The x- coordinate of the vertex is x = - b 2 a = - 2 2(1) = - 2 2 = - 1. The y- coordinate of the vertex is f - b 2 a  = f ( - 1) = ( - 1) 2 + 2( - 1) = 1 - 2 = - 1.
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