# 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).