Section 5.1
The Parabola
419
Version: Fall 2007
5.1
The Parabola
In this section you will learn how to draw the graph of the quadratic function defined
by the equation
f
(
x
) =
a
(
x
−
h
)
2
+
k.
(1)
You will quickly learn that the graph of the quadratic function is shaped like a "U"
and is called a
parabola
. The form of the quadratic function in
equation (1)
is called
vertex form
, so named because the form easily reveals the
vertex
or “turning point”
of the parabola.
Each of the constants in the vertex form of the quadratic function
plays a role. As you will soon see, the constant
a
controls the scaling (stretching or
compressing of the parabola), the constant
h
controls a horizontal shift and placement
of the
axis of symmetry
, and the constant
k
controls the vertical shift.
Let’s begin by looking at the
scaling
of the quadratic.
Scaling the Quadratic
The graph of the basic quadratic function
f
(
x
) =
x
2
shown in
Figure 1
(a) is called
a
parabola
. We say that the parabola in
Figure 1
(a) “opens upward.” The point at
(0
,
0)
, the “turning point” of the parabola, is called the
vertex
of the parabola. We’ve
tabulated a few points for reference in the table in
Figure 1
(b) and then superimposed
these points on the graph of
f
(
x
) =
x
2
in
Figure 1
(a).
x
y
5
5
f
x
f
(
x
) =
x
2
−
2
4
−
1
1
0
0
1
1
2
4
(a) A basic parabola.
(b) Table of
x
values
and function values
satisfying
f
(
x
) =
x
2
.
Figure 1.
The graph of the basic
parabola is a fundamental starting point.
Now that we know the basic shape of the parabola determined by
f
(
x
) =
x
2
, let’s
see what happens when we scale the graph of
f
(
x
) =
x
2
in the vertical direction. For
Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
420
Chapter 5
Quadratic Functions
Version: Fall 2007
example, let’s investigate the graph of
g
(
x
) = 2
x
2
.
The factor of 2 has a doubling
effect. Note that each of the function values of
g
is twice the corresponding function
value of
f
in the table in
Figure 2
(b).
x
y
5
10
f
g
x
f
(
x
) =
x
2
g
(
x
) = 2
x
2
−
2
4
8
−
1
1
2
0
0
0
1
1
2
2
4
8
(a) The graphs of
f
and
g
.
(b) Table of
x
values and function values
satisfying
f
(
x
) =
x
2
and
g
(
x
) = 2
x
2
.
Figure 2.
A stretch by a factor of 2 in the vertical direction.
When the points in the table in
Figure 2
(b) are added to the coordinate system in
Figure 2
(a), the resulting graph of
g
is stretched by a factor of two in the vertical
direction.
It’s as if we had put the original graph of
f
on a sheet of rubber graph
paper, grabbed the top and bottom edges of the sheet, and then pulled each edge in
the vertical direction to stretch the graph of
f
by a factor of two. Consequently, the
graph of
g
(
x
) = 2
x
2
appears somewhat narrower in appearance, as seen in comparison
to the graph of
f
(
x
) =
x
2
in
Figure 2
(a). Note, however, that the vertex at the origin
is unaffected by this scaling.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Quadratic equation, 5 g, 5 g, 1 0 1 2 g, 1 0 1 g

Click to edit the document details