is the
minimum value
of
f
.
The parabola opens down if
; the vertex is a maximum point and
f

b
2
a
!
is the
maximum value
of
f
.
56
The
y
intercept is the value of
f
at
x
= 0. That is, the
y
intercept is
The
x
intercepts, if any, are found by solving the quadratic equation
ax
2
+
bx
+
c
= 0.
Recall the
Quadratic Formula
.
x
=
If the discriminant
b
2

4
ac >
0, the graph has
If the discriminant
b
2

4
ac
= 0, the graph has
If the discriminant
b
2

4
ac
= 0, the graph will touch the
x
axis at its
If the discriminant
b
2

4
ac <
0, the graph has
When the quadratic function is in the form:
f
(
x
) =
a
(
x

h
)
2
+
k
, we can use what we
learned about
transformations
in Sec 3.5 to identify the vertex.
h
is the
shift and
k
is the
shift.
The
vertex
is
.
In this form, it is still true that if
a >
0 the parabola opens
and
if
a <
0 the parabola opens
.
Recall, again,
Section 2.2
.
The
y
intercept is the value of
f
at
x
= 0. That is, the
y
intercept =
f
(0).
You must calculate it.
The
x
intercepts, if any, are found by letting
y
= 0. That is, the
x
intercepts, if any, are found by
solving the quadratic equation
a
(
x

h
)
2
+
k
= 0.
Recall
Section 1.2
.
This equation is best solved by the
Squareroot Property
.
Do not
.
***************
Summary
Standard Form
(
h, k
) Form
f
(
x
) =
ax
2
+
bx
+
c
f
(
x
) =
a
(
x

h
)
2
+
k
vertex

b
2
a
, f

b
2
a
!!
(
h, k
)
a
a >
0 opens up
same
a <
0 opens down
same
y
int
let
x
= 0, solve for
y
same
y
int=
c
calculate the
y
intercept
x
int(s)
let
y
= 0, solve for
x
same
(factor or quad. formula)
(use squareroot property)
57
Match each function with one of the graphs below.
f
(
x
) =

x
2

1
f
(
x
) = (
x
+ 1)
2

1
f
(
x
) =
x
2

2
x
f
(
x
) =
x
2
+ 2
x
+ 1
f
(
x
) =
x
2
+ 2
x
+ 2
x
y
1
x
y
1
1
1
1
x
y
1
x
y
1
x
y
1
58
Graph the function
f
(
x
) = (
x

3)
2

10
by starting with the graph of
y
=
x
2
and using
transformations (shifting, compressing, stretching, and/or reflecting.)
x
y
Graph each quadratic function by determining whether its graph opens up or down and
by finding its vertex, axis of symmetry,
y
intercept, and any
x
intercepts. Determine
the domain and range.
Determine where the function is increasing and where it is
decreasing.
f
(
x
) =

x
2
+4
x
f
(
x
) = (
x
+2)
2

4
f
(
x
) =
x
2
+6
x
+9
x
y
x
y
x
y
59
f
(
x
) =

3
x
2
+ 3
x

2
f
(
x
) = 2
x
2
+ 5
x
+ 3
x
y
x
y
Determine, without graphing, whether the given quadratic function has a maximum
value or a minimum value, and then find the value.
f
(
x
) = 4
x
2

8
x
+ 3
f
(
x
) =

3
x
2
+ 12
x
+ 1
f
(
x
) = (
x

3)
2
+ 5
f
(
x
) =

2(
x
+ 3)
2

7
Determine, without graphing, the interval on which the given function is increasing
and on which it is decreasing.
f
(
x
) = 4
x
2

8
x
+ 3
f
(
x
) =

3
x
2
+ 12
x
+ 1
f
(
x
) = (
x

3)
2
+ 5
f
(
x
) =

2(
x
+ 3)
2

7
60
The John Deere company has found that the revenue, in dollars, from sales of riding
mowers is a function of the unit price
p
, in dollars, that it charges.
If the revenue
R
is
R
(
p
) =

1
2
p
2
+ 1900
p
what unit price
p
should be charged to maximize revenue? What is the
maximum revenue?
The marginal cost
C
(in dollars) of manufacturing
x
cell phones (in thousands) is given
by
C
(
x
) = 5
x
2

200
x
+ 4000
.
How many cell phones should be manufactured to minimize the
marginal cost? What is the minimal marginal cost?