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.
Graph the quadratic inequality.
y x2 – x – 12
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)
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).