# ( ) EXAMPLE 6 Write a quadratic function in vertex form

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( ) 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. Add –10 2 ( ) = (–5) 25 to each side. 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. The vertex form of the function is y = (x – 5)2 – 3. The vertex is (5, – 3). ANSWER

EXAMPLE 7 Find the maximum value of a quadratic function Baseball The height y (in feet) of a baseball t seconds 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.

Find the maximum value of a quadratic function
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 Add to each side. (–16)(9) y – 144 = –16(t – 3)2 + 3 Write t2 – 6t + 9 as a binomial squared. y = –16(t – 3) Solve for y. The vertex is (3, 147), so the maximum height of the baseball is 147 feet. ANSWER

( ) GUIDED PRACTICE for Examples 6 and 7 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. Add –8 2 ( ) = (–4) 16 to each side. 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. The vertex form of the function is y = (x – 4) The vertex is (4, 1). ANSWER

( ) GUIDED PRACTICE for Examples 6 and 7 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. Add 6 2 ( ) = (3) 9 to each side. 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. The vertex form of the function is y = (x + 3)2 – 6. The vertex is (– 3, – 6). ANSWER

( ) GUIDED PRACTICE for Examples 6 and 7 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. Add – 4 2 ( ) = (– 2) 4 to each side. y + 4 = (x2 – 4x + 4) – 4 y + 4 = (x – 2)2 – 4 Write x2 – 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

GUIDED PRACTICE for Examples 6 and 7 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.

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