Graphing Quadratic Inequalities

To graph the quadratic inequality, we will follow three steps.

Graph the related quadratic function.
Test a point not on the parabola.
Shade accordingly. If the point satisfies the inequality, we shade the region that contains the point. If not, we shade the opposite region.

Step $1$

Let's draw the graph of the related function, which is $y = x^{2} + 5 x + 6 .$

Step $2$

Next, let's determine which region to shade by testing a point. For simplicity, we will use

(0, 0)

as our test point. Let's see if it satisfies the given inequality.

y \geq x^{2} + 5 x + 6

SubstituteII

x = 0

,

y = 0

0 \geq ? (0)^{2} + 5 (0) + 6

CalcPowCalculate power

0 \geq ? (0) + 5 (0) + 6

ZeroPropMultZero Property of Multiplication

0 \geq ? 0 + 0 + 6

IdPropAddIdentity Property of Addition

0 ≱ 6 \times

Step $3$

Since $(0, 0)$ did not produce a true statement, we will shade the region that does not contain the point. Also, note that the inequality is not strict. Therefore, the parabola will be solid.