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EXAMPLE 6

). (. –10. Add. to each side. 2. 25. (–5). 2. =. =. 2. ANSWER. The vertex form of the function is y = ( x – 5) 2 – 3 . The vertex is (5, – 3). EXAMPLE 6. Write a quadratic function in vertex form. Write y = x 2 – 10 x + 22 in vertex form. Then identify the vertex.

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EXAMPLE 6

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  1. ) ( –10 Add to each side. 2 25 (–5) 2 = = 2 ANSWER The vertex form of the function is y = (x – 5)2– 3. The vertex is (5, – 3). EXAMPLE 6 Write a quadratic function in vertex form Write y = x2 – 10x + 22 in vertex form. Then identify the vertex. y = x2 – 10x + 22 Write original function. y + ?= (x2–10x + ?) + 22 Prepare to complete the square. y + 25= (x2– 10x + 25) + 22 y + 25 = (x – 5)2 + 22 Write x2 – 10x + 25 as a binomial squared. y = (x – 5)2– 3 Solve for y.

  2. EXAMPLE 7 Find the maximum value of a quadratic function Baseball The height y(in feet) of a baseball tseconds after it is hit is given by this function: y = –16t2 + 96t + 3 Find the maximum height of the baseball. SOLUTION The maximum height of the baseball is the y-coordinate of the vertex of the parabola with the given equation.

  3. (–16)(9) Add to each side. ANSWER The vertex is (3, 147), so the maximum height of the baseball is 147feet. EXAMPLE 7 Find the maximum value of a quadratic function y = – 16t2+ 96t +3 Write original function. y = – 16(t2– 6t)+3 Factor –16 from first two terms. y +(–16)(?)= –16(t2 –6t + ?) + 3 Prepare to complete the square. y +(–16)(9)= –16(t2 –6t + 9) + 3 y – 144 = –16(t – 3)2 + 3 Write t2 – 6t + 9 as a binomial squared. y = –16(t – 3)2+ 147 Solve for y.

  4. ) ( –8 Add to each side. 2 16 (–4) 2 = = 2 ANSWER The vertex form of the function is y = (x – 4)2+ 1. The vertex is (4, 1). for Examples 6 and 7 GUIDED PRACTICE 13. Write y = x2 – 8x + 17 in vertex form. Then identify the vertex. y = x2 – 8x + 17 Write original function. y + ?= (x2–8x + ?) + 17 Prepare to complete the square. y + 16= (x2– 8x + 16) + 17 Write x2 – 8x + 16 as a binomial squared. y + 16 = (x – 4)2 + 17 y = (x – 4)2+ 1 Solve for y.

  5. ) ( 6 Add to each side. 2 9 (3) 2 = = 2 ANSWER The vertex form of the function is y = (x + 3)2– 6. The vertex is (– 3, – 6). for Examples 6 and 7 GUIDED PRACTICE 14. Write y = x2 + 6x + 3 in vertex form. Then identify the vertex. y = x2 + 6x + 3 Write original function. y + ?= (x2+ 6x + ?) + 3 Prepare to complete the square. y + 9= (x2+ 6x + 9) + 3 Write x2 + 6x + 9 as a binomial squared. y + 9 = (x + 3)2 + 3 y = (x + 3)2– 6 Solve for y.

  6. ) ( – 4 Add to each side. 2 4 (– 2) 2 = = 2 ANSWER The vertex form of the function is y = (x – 2)2– 8. The vertex is (2 , – 8). for Examples 6 and 7 GUIDED PRACTICE 15. Write f(x) = x2 – 4x – 4 in vertex form. Then identify the vertex. f(x) = x2 – 4x – 4 Write original function. y + ?= (x2– 4x + ?) – 4 Prepare to complete the square. y + 4= (x2–4x + 4) – 4 Write x2 – 4x + 4 as a binomial squared. y + 4 = (x – 2)2 – 4 y = (x – 2)2– 8 Solve for y.

  7. for Examples 6 and 7 GUIDED PRACTICE 16. What if ? In example 7, suppose the height of the baseball is given by y = – 16t2 + 80t + 2. Find the maximum height of the baseball. SOLUTION The maximum height of the baseball is the y-coordinate of the vertex of the parabola with the given equation. y = – 16t2+ 80t +2 Write original function. y = – 4((2t)2– 20t)+2 Factor – 4 from first two terms. y +(– 4)(?)= – 4((2t)2 – 20t + ?) + 2 Prepare to complete the square.

  8. (–4)(25) Add to each side. ANSWER The vertex is (5, 102), so the maximum height of the baseball is 102feet. for Examples 6 and 7 GUIDED PRACTICE y +(– 4)(25)= – 4((2t)2– 20t + 25) + 2 Write 2t2 – 20+ 25 as a binomial squared. y – 100 = – 4(2t – 5)2 + 2 y = – 4(2t – 5)2+ 102 Solve for y.

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