4.1
Quadratic Functions
381
EXAMPLE 3
Graphing a Quadratic Function Given in
General Form
Graph the quadratic function
f
(
x
)
x
2
6
x
4.
Solution:
Express the quadratic function in standard form by completing the square.
Write the original function.
f
(
x
)
x
2
6
x
4
Group the variable terms together.
(
x
2
6
x
)
4
Complete the square.
Half of
6 is
3;
3 squared is 9.
Add and subtract 9 within the parentheses.
(
x
2
6
x
9
9
)
4
Write the
9 outside the parentheses.
(
x
2
6
x
9)
9
4
Write the expression inside the parentheses as
a perfect square and simplify.
(
x
3)
2
5
Now that the quadratic function is written in standard form, f
(
x
)
(
x
3)
2
5,
we follow our stepbystep procedure for graphing a quadratic function in standard
form.
S
TEP
1
The parabola opens up.
a
1, so
a
0
S
TEP
2
Determine the vertex.
(
h
,
k
)
(3,
5)
S
TEP
3
Find the
y
intercept.
f
(0)
(0)
2
6(0)
4
4
(0, 4) corresponds to the
y
intercept
S
TEP
4
Find any
x
intercepts.
f
(
x
)
0
f
(
x
)
(
x
3)
2
5
0
(
x
3)
2
5
(
, 0
)
and
(
, 0
)
correspond to the
x
intercepts.
S
TEP
5
Plot the vertex and intercepts
(3,
5), (0, 4),
(
, 0
)
,
and
(
, 0
)
.
Connect the points with a smooth parabolic
curve.
Note:
and
■
YOUR TURN
Graph the quadratic function
f
(
x
)
x
2
8
x
14.
3

1
5
L
0.76.
3
+
1
5
L
5.24
3

1
5
3
+
1
5
3

1
5
3
+
1
5
x
=
3
;
1
5
x

3
= ;
1
5
Study Tip
Although either form (standard or
general) can be used to find the inter
cepts, it is often more convenient to
use the general form when finding
the
y
intercept and the standard form
when finding the
x
intercept.
Technology Tip
Use a graphing utility to graph the
function
f
(
x
)
x
2
6
x
4 as
y
1
.
x
y
(3, –5)
(0, 4)
(0.76, 0)
(5.24, 0)
■
Answer:
x
y
(5.4, 0)
(2.6, 0)
(4, –2)
Typically, quadratic functions are expressed in general form and a graph is the ultimate
goal, so we must first express the quadratic function in standard form. One technique for
transforming a quadratic function from general form to standard form was introduced in
Section 1.3 and is called
completing the square.
382
CHAPTER 4
Polynomial and Rational Functions
EXAMPLE 4
Graphing a Quadratic Function Given in General Form
with a Negative Leading Coefficient
Graph the quadratic function
f
(
x
)
3
x
2
6
x
2.
Solution:
Express the function in standard form by completing the square.
Write the original function.
f
(
x
)
3
x
2
6
x
2
Group the variable terms together.
(
3
x
2
6
x
)
2
Factor out
3 in order to make the coefficient
of
x
2
equal to 1 inside the parentheses.
3
(
x
2
2
x
)
2
Add and subtract 1 inside the
parentheses to create a perfect square.
3
(
x
2
2
x
1
1
)
2
Regroup the terms.
3
(
x
2
2
x
1
)
3(
1
)
2
Write the expression inside the parentheses
as a perfect square and simplify.
3(
x
1)
2
5
Now that the quadratic function is written in standard form, f
(
x
)
3(
x
1)
2
5,
we follow our stepbystep procedure for graphing a quadratic function in standard form.
S
TEP
1
The parabola opens down.
a
3, therefore
a
0
S
TEP
2
Determine the vertex.
(
h
,
k
)
(1, 5)
S
TEP
3
Find the
y
intercept using
f
(0)
3(0)
2
6(0)
2
2
the general form.
(0, 2) corresponds to the
y
intercept
S
TEP
4
Find any
x
intercepts using
the standard form.
f
(
x
)
3(
x
1)
2
5
0
3(
x
1)
2
5
x

1
= ;
A
5
3
(
x

1)
2
=
5
3
x
y
(1, 5)
(0, 2)
(–0.3, 0)
(2.3, 0)
■
Answer:
x
y
–5
5
5
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 Summer '17
 juan alberto
 Completing The Square, Quadratic equation