# EXAMPLE 6 Write a quadratic function in vertex

- Slides: 8

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

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 = – 16 t 2 + 96 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.

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

GUIDED PRACTICE for Examples 6 and 7 13. Write y = x 2 – 8 x + 17 in vertex form. Then identify the vertex. y = x 2 – 8 x + 17 Write original function. y + ? = (x 2 – 8 x + ? ) + 17 Prepare to complete the square. y + 16 = (x 2 – 8 x + 16) + y + 16 = (x – 4)2 + 17 y = (x – 4)2 + 1 ( ) 2 17 Add – 82 = (– 4)2= 16 to each side. Write x 2 – 8 x + 16 as a binomial squared. Solve for y. ANSWER The vertex form of the function is y = (x – 4)2 + 1. The vertex is (4, 1).

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

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

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

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