Solution
The only difference between the two functions is the value of
c
.
Since the value of
c
is
increased by 4, the graph of
( )
g x
can be found by vertically shifting the graph of
( )
f
x
up 4 units.
In order to compare two quadratic functions where the values of
a
,
b
, and/or
c
change, it is easier to look
at the
standard form
of a quadratic function.
By completing the square we get
2
2
2
2
2
2
2
2
2
2
2
2
4
2
4
2
( )
(
)
(
(
) )
(
)
(
)
(
)
(
(
))
(
)
(
)
b
b
b
b
a
a
a
a
b
b
b
b
a
a
a
a
f x
ax
bx
c
a x
x
c
a x
x
c
a
a x
c
a x
c
a x
h
k
where
2
b
a
h
and
2
4
( )
b
a
k
c
f h
.
In standard form, the graph of the parabola can be found by
shifting the graph of
2
( )
f
x
x
horizontally (right if
0
h
, left if
0
h
) by
h
units, vertically expanding
or contracting by a factor of
a
, vertically reflecting if
0
a
, and then shifting vertically by
k
units.
Notice that by performing this transformation, the graph of a quadratic in standard form has a vertex at
the point
( , )
h k
.
Table 1.1
Summary Chart for Parabolas
Let
2
( )
(
)
f
x
a x
h
k
.
Comparing this to the parabola
2
x
,
1.
If
0
a
then
( )
f
x
opens upward and the vertex is the lowest point.
2.
If
0
a
then
( )
f
x
vertically reflected and the vertex is the highest point.
3.
If
0
1
a
then
( )
f
x
is vertically compressed relative to
2
x
.
4.
If
1
a
then
( )
f
x
is vertically expanded relative to
2
x
.
5.
If
0
k
, then
( )
f
x
is vertically shifted upwards by
k
units.
6.
If
0
k
, then
( )
f
x
is vertically shifted downwards by
k
units.
7.
If
0
h
, then
( )
f
x
is horizontally shifted to the right by
h
units.
8.
If
0
h
, then
( )
f
x
is horizontally shifted to the left by
h
units.
9.
The vertex is at
( , )
h k
.
Example 1.13
Find the vertex of the parabola
2
( )
2
3
6
f
x
x
x
and determine whether it is a
maximum or minimum.
Solution
The
x
-coordinate of the vertex is given by
( 3)
0.75
2
2(2)
b
x
a
The
y
-coordinate of the vertex can be found by plugging in the corresponding
x
value:
2
(0.75)
2(0.75)
3(0.75)
6
4.875
f

Therefore, the vertex of the parabola is located at (0.75, 4.875).
Since
2
0
a
, then the parabola opens
upward from the vertex, so the vertex is a minimum.
Example 1.14
How is the graph of
2
1
5
( )
(
2)
4
f
x
x
different from the graph of
2
( )
g x
x
?
Solution
The graph will be horizontally shifted to the left by 2 units, reflected over the
x
-axis,
vertically contracted by a factor of
1
5
, and then vertically shifted down 4 units.
Using the Plotting
Applet, these transformations can be seen in Figure 1.14.
Figure 1.13
Graphs for Example 1.11 using the Plotting Applet
Many real-life situations are quadratic in nature, as they portray information that increases to a point and
then decreases, such as revenue or profit, or decreases to a point and then increases, such as costs.

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- Fall '16
- rioj
- Supply And Demand, Quadratic equation