#
Section 9-3

#
Quadratic Functions

The general form of a quadratic function is
*
***
f
**
**
(
***
x
*
) =
*
ax
*
^{
2
}
+
*
bx
*
+
*
c
*
. The graph of such a function is a
parabola
, a type of 2-dimensional curve.

The "basic" parabola,
*
***
y
**
**
=
***
x
*
^{
2
}
, looks like this:

The function of the coefficient
*
***
a
**
in the general equation is to make the parabola "wider" or "skinnier", or to turn it upside down (if negative):

If the coefficient of
*
x
*
^{
2
}
is positive, the parabola opens up; otherwise it opens down.

##
THE VERTEX

The
vertex of a parabola
is the point at the bottom of the "U" shape (or the top, if the parabola opens downward).

The equation for a parabola can also be written in "vertex form":

*
***
f
**
**
(
***
x
*
) =
*
a
*
(
*
x
*
–
**
***
h
*
)
^{
2
}
+
*
k
*

In this equation, the vertex of the parabola is the point (
*
h
*
,
*
k
*
).

You can see how this relates to the standard equation by multiplying it out:

*
***
f
**
**
(
***
x
*
) =
*
a
*
(
*
x
*
–
**
***
h
*
)(
*
x
*
–
**
***
h
*
) +
*
k
*

*
***
f
**
**
(
***
x
*
) =
*
a
*
*
x
*
^{
2
}
–
**
2
***
a
*
*
hx
*
+
*
a
*
*
h
*
^{
2
}
+
*
k
*

The coefficient of
*
x
*
here is –2
*
a
*
*
h
*
. This means that in the standard form,
*
f
*
(
*
x
*
) =
**
***
ax
*
^{
2
}
+
*
bx
*
+
*
c
*
, the expression

gives the
*
x
*
-coordinate of the vertex
*
.
*

Example:

Find the vertex of the parabola.

*
***
f
**
**
(
***
x
*
) = 3
*
x
*
^{
2
}
+ 12
*
x
*
–
**
12
**

Here,
*
a
*
= 3 and
*
b
*
= 12. So, the
*
x
*
-coordinate of the vertex is:

Substituting in the original equation to get the
*
y
*
-coordinate, we get:

*
***
f
**
**
(
**
–
**
2) = 3(
**
–
**
2)
**^{
2
}
+ 12(
–
**
2)
**
–
**
12
**

**
=
**
–
**
24
**

So, the vertex of the parabola is at (–2, –24).

##
AXIS OF SYMMETRY

The
axis of symmetry
of a parabola is the vertical line through the vertex. For a parabola in standard form,
**
***
***
f
**
**
(
***
x
*
) =
*
ax
*
^{
2
}
+
*
bx
*
+
*
c
*
, the axis of symmetry has the equation

Example:

Find the axis of symmetry.

*
***
f
**
**
(
***
x
*
) = 2
*
x
*
^{
2
}
+
*
x
*
–
**
1
**

Here,
*
a
*
= 2 and
*
b
*
= 1. So, the axis of symmetry is the vertical line

##
DOMAIN AND RANGE

As with any function, the
domain
**
**
of a quadratic function
*
f
*
(
*
x
*
) is the set of
*
x
*
-values for which the function is defined, and the
range
is the set of all the output values (values of
*
f
*
).

Parabolas generally have the whole real line as their domain: any
*
x
*
is a legitimate input. The range is restricted to those points greater than or equal to the
*
y
*
-coordinate of the vertex (or less than or equal to, depending on whether the parabola opens up or down).