Graphing Quadratic Inequalities

A quadratic inequality of the form

y ax² + 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.

The related equation is:

y = x² – x – 12

First we notice that a, the coefficient of the x² 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).