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Published byJayde Titcomb Modified about 1 year ago

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

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EXAMPLE 7 Find the maximum value of a quadratic function The height y (in feet) of a baseball t seconds after it is hit is given by this function: Baseball y = –16t t + 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.

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

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

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

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

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

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

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