3.1 Quadratic Functions Examples And Worksheet - Chapter 3: Polynomial And Rational Functions Page 2
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Section 3.1
Quadratic Functions
Library of Parent Functions: Quadratic Function
2
The simplest type of quadratic function is
f x
ax
,
also known as the
squaring function when
a
1.
The basic characteristics of a quadratic
function are summarized below. A review of quadratic functions can be
found in the Study Capsules.
2
2
, a > 0
, a < 0
Graph of
f x
ax
Graph of
f x
ax
Domain:
,
Domain:
,
Range:
0,
Range:
, 0
Intercept:
0, 0
Intercept:
0, 0
Decreasing on
, 0
Increasing on
, 0
Increasing on
0,
Decreasing on
0,
Even function
Even function
Axis of symmetry:
x
0
Axis of symmetry:
x
0
Relative minimum or vertex:
0, 0
Relative maximum or vertex:
0, 0
f(x) = ax , a > 0
2
y
y
3
2
2
1
Maximum: (0, 0)
1
x
−3 −2 −1
1
2
3
−1
x
−3 −2 −1
f(x) = ax , a < 0
2
1
2
3
−1
−2
Minimum: (0, 0)
−2
−3
2
For the general quadratic form
f x
ax
bx
c,
if the leading
coefficient is positive, the parabola opens upward; and if the leading
a
coefficient is negative, the parabola opens downward. Later in this section
a
you will learn ways to find the coordinates of the vertex of a parabola.
y
y
Opens upward
2
f x
( ) =
ax
+
bx
+
c, a <
0
Vertex is
Axis
highest point
Axis
Vertex is
>
2
f x
( ) =
ax
+
bx
+
c, a
0
lowest point
Opens
x
x
downward
2
When sketching the graph of
f x
ax
,
it is helpful to use the graph of
2
y
x
as a reference, as discussed in Section 1.5. There you saw that when
a > 1,
the graph of
y
af x
is a vertical stretch of the graph of
y
f x .
When
0 < a < 1,
the graph of
y
af x
is a vertical shrink of the graph of
y
f x .
This is demonstrated again in Example 1.
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