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Graphing Quadratic Inequalities

A quadratic inequality of the form

y > a x 2 + b x + 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".

Math diagram

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:

0 ? 0 2 0 12 0 12

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

Math diagram

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