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
y + ? = (x2 –10x + ? ) + 22
y + 25 = (x2 – 10x + 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 x2 – 10x + 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 = –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.
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 (–16)(9) to each side.
y – 144 = –16(t – 3)2 + 3
y = –16(t – 3)2 + 147
Write t2 – 6t + 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 = 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.
2
y + 16 =
– 8x + 16) + 17Add –82 = (–4)2= 16 to each side.
Write x2 – 8x + 16 as a
y + 16 = (x – 4)2 + 17
binomial squared.
(x2
y = (x – 4)2 + 1
( )
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 = 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
y + 9 = (x + 3)2 + 3
y = (x + 3)2 – 6
2
( )
2
6
Add
(3)
= 9 to each side.
2 =
Write x2 + 6x + 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) = 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
y + 4 = (x – 2)2 – 4
y = (x – 2)2 – 8
2
2
Add – 4 = (– 2) = 4 to each side.
2
Write x2 – 4x + 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 = – 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 (–4)(25) to each side.
y – 100 = – 4(2t –
5)2
+2
y = – 4(2t – 5)2 + 102
Write 2t2 – 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.