#
Section 10-5

#
Graphing Quadratic Inequalities

A quadratic
inequality
of the form

*
***
y
**
**
>
***
ax
*
^{
2
}
+
*
bx
*
+
*
c
*

(or substitute <,
, or
for >) represents a region of the plane bounded by a
parabola
.

To graph a
quadratic inequality
, start by graphing the parabola. Then fill in the region either above or below it, depending on the inequality.

If the inequality symbol is
or
, then the region includes the parabola, so it should be graphed with a solid line.

Otherwise, if the inequality symbol is < or >, the parabola should be drawn with a dotted line to indicate that the region does not include its boundary.

Example:

Graph the quadratic inequality.

*
***
y
**
**
***
x
*
^{
2
}
–
**
***
x
*
–
**
12
**

The related equation is:

*
***
y
**
**
=
***
x
*
^{
2
}
–
**
***
x
*
–
**
12
**

First we notice that
*
a
*
, the coefficient of the
*
x
*
^{
2
}
term, is equal to 1. Since
*
a
*
is positive, the parabola points upward.

The right side can be factored as:

**
***
y
*
= (
*
x
*
+ 3)(
*
x
*
–
**
4)
**

So the parabola has
*
x
*
-intercepts
at –3 and 4. The vertex must lie midway between these, so the
*
x
*
-coordinate of the vertex is 0.5.

Plugging in this
*
x
*
-value, we get:

**
***
y
*
= (0.5 + 3)(0.5
–
**
4)
**

*
***
y
**
**
= (3.5)(
**
–
**
3.5)
**

*
***
y
**
**
=
**
–
**
12.25
**

So, the vertex is at (0.5, –12.25).

We now have enough information to graph the parabola. Remember to graph it with a solid line, since the inequality is "less than or equal to".

Should you shade the region inside or outside the parabola? The best way to tell is to plug in a sample point. (0, 0) is usually easiest:

So, shade the region which does
**
not
**
include the point (0, 0).