Alg. 2 Chapter 5.5 - Beau Chene High School Home Page

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GOAL 2 WRITING QUADRATIC FUNCTIONS IN VERTEX FORM Given a quadratic function in standard form, y = ax 2 + bx + c, you can use completing the square to write the function in vertex form, y = a(x º h) 2 + k. EXAMPLE 6 Writing a Quadratic Function in Vertex Form STUDENT HELP HOMEWORK HELP Visit our Web site www.mcdougallittell.com for extra examples. INTERNET Write the quadratic function y = x 2 º8x + 11 in vertex form. What is the vertex of the function’s graph? SOLUTION y = x 2 º8x +11 Write original function. y + ? = (x 2 º8x + ? ) + 11 Prepare to complete the square for x 2 º 8x. y + 16 = (x 2 º8x + 16)+11 y +16= (x º4) 2 +11 º8 } 2 Add 2 = (º4) 2 = 16 to each side. Write x 2 º 8x + 16 as a binomial squared. y = (x º4) 2 º5 Solve for y. The vertex form of the function is y = (x º4) 2 º 5. The vertex is (4, º5). REAL LIFE EXAMPLE 7 Finding the Maximum Value of a Quadratic Function Agriculture The amount s (in pounds per acre) of sugar produced from sugarbeets can be modeled by the function s = º0.0655n 2 + 7.855n + 5562 where n is the amount (in pounds per acre) of nitrogen fertilizer used. How much fertilizer should you use to maximize sugar production? What is the maximum amount of sugar you can produce? Source: Sugarbeet Research and Education Board of Minnesota and North Dakota SOLUTION The optimal amount of fertilizer and the maximum amount of sugar are the coordinates of the vertex of the function’s graph. One way to find the vertex is to write the function in vertex form. s = º0.0655n 2 + 7.855n + 5562 s = º0.0655(n 2 º 120n) + 5562 s º 0.0655( ? ) = º0.0655(n 2 º 120n + ? ) + 5562 s º 0.0655(3600) = º0.0655(n 2 º 120n + 3600) + 5562 s º 236 = º0.0655(n º 60) 2 + 5562 s = º0.0655(n º 60) 2 + 5798 The vertex is approximately (60, 5798). To maximize sugar production, you should use about 60 pounds per acre of nitrogen fertilizer. The maximum amount of sugar you can produce is about 5800 pounds per acre. <strong>5.5</strong> Completing the Square 285
GUIDED PRACTICE Vocabulary Check ✓ Concept Check ✓ Skill Check ✓ 1. Describe what it means to “complete the square” for an expression of the form x 2 + bx. 2. Which method for solving quadratic equations—factoring or completing the square—is more general? Explain. 3. ERROR ANALYSIS A student tried to write y = ºx 2 º6x + 4 in vertex form as shown. Explain the student’s mistake. Then write the correct vertex form of the function. Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. 4. x 2 +2x + c 5. x 2 +14x + c 6. x 2 º6x + c 7. x 2 º10x + c 8. x 2 +5x + c 9. x 2 º13x + c Solve the equation by completing the square. y =ºx 2 º 6x + 4 y =º(x 2 + 6x) + 4 y + 9 =º(x 2 + 6x + 9) + 4 y + 9 =º(x + 3) 2 + 4 y =º(x + 3) 2 º 5 10. x 2 +4x =º1 11. x 2 º2x =4 12. x 2 º16x +76=0 13. x 2 +8x +9=0 14. 2x 2 +12x =4 15. 3x 2 º12x + 93 = 0 Write the quadratic function in vertex form and identify the vertex. 16. y = x 2 +12x 17. y = x 2 º4x +7 18. y = x 2 º8x +31 19. y = x 2 +10x +17 20. y =ºx 2 +14x º45 21. y =2x 2 +4x º4 22. LANDSCAPE DESIGN Suppose the homeowner in Example 5 has 60 feet of wire mesh to put around the garden and enough mulch to cover an area of 140 square feet. What should the dimensions of the garden be? PRACTICE AND APPLICATIONS STUDENT HELP Extra Practice to help you master skills is on p. 946. REWRITING EXPRESSIONS Write the expression as the square of a binomial. 23. x 2 +16x +64 24. x 2 +20x + 100 25. x 2 º24x + 144 26. x 2 º38x + 361 27. x 2 + x +0.25 28. x 2 º1.4x +0.49 29. x 2 º3x + 9 4 30. x2 + 1 6 x + 1 144 31. x2 º 4 9 x + 4 81 COMPLETING THE SQUARE Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. 32. x 2 º12x + c 33. x 2 +18x + c 34. x 2 +26x + c 35. x 2 º44x + c 36. x 2 +9x + c 37. x 2 º11x + c 38. x 2 º23x + c 39. x 2 +15x + c 40. x 2 º0.2x + c 41. x 2 º5.8x + c 42. x 2 +1.6x + c 43. x 2 +9.4x + c 44. x 2 º 2 7 x + c 45. x2 + 1 0 x + c 46. x 3 2 + 1 7 x + c 8 286 <strong>Chapter</strong> 5 Quadratic Functions
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Alg. 2 Chapter 5.5 - Beau Chene High School Home Page

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