Transcript y - ClassZone
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
Write original function.
Prepare to complete the square.
Add
2 ( ) = (–5) 2 = 25
Write
x
2 – 10
x
+ 25
to each side.
as a binomial squared.
y
= (
x
– 5) 2 – 3
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
y
= – 16(
t
2 – 6
t
) +3
y
+(–16)( ?
) = –16(
t
2 –6
t
+ ?
) + 3
y
+(–16)( 9 ) = –16(
t
2 –6
t
+ 9 ) + 3
y
– 144 = –16(
t
– 3) 2 + 3
Write original function.
Factor
–16
from first two terms.
Prepare to complete the square.
Add
(–16)(9)
to each side.
Write
t
2 – 6
t
+ 9
as a binomial squared.
y
= –16(
t
– 3) 2 + 147
Solve for
y
.
ANSWER The verte
x
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) + 17
Add
y
+ 16 = (
x
– 4) 2 + 17 2 ( ) = (–4) 2 = 16
Write
x
2 – 8
x
+ 16
as a binomial squared.
to each side.
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
=
x
2 + 6
x
+ 3
in vertex form. Then identify the vertex.
y
=
x
2 + 6
x
+ 3
y
+ ? = (
x
2 + 6
x
+ ? ) + 3
y
+ 9 = (
x
2 + 6
x
+ 9) + 3
y
+ 9 = (
x
+ 3) 2 + 3
y
= (
x
+ 3) 2 – 6
Write original function.
Prepare to complete the square.
Add
2 ( ) = (3) 2 = 9
to each side.
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
y
+ ? = (
x
2 – 4
x
+ ? ) – 4
y
+ 4 = (
x
2 – 4
x
+ 4) – 4
y
+ 4 = (
x
– 2) 2 – 4
Write original function.
Prepare to complete the square.
Add
2 ( ) = (– 2) 2
Write
x
2 – 4
x
+ 4 = 4
as a to each side.
binomial squared.
y
= (
x
– 2) 2 – 8
Solve for
y
.
ANSWER The vertex form of the function is
y
= (
x
– 2) 2 – 8
. The vertex is
(2 , – 8).
GUIDED PRACTICE
for Examples 6 and 7 16.
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
y
+(– 4)(?) = – 4((2
t
) 2 – 20
t
+ ? ) + 2
Factor
– 4
from first two terms.
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
Write
2
t
2 – 20 + 25
binomial squared.
as a
y
= – 4(2
t
– 5) 2 + 102
Solve for
y
.
ANSWER The vertex is
(5, 102)
, so the maximum height of the baseball is
102 feet
.