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Graphing Quadratic Inequalities

We've graphed quadratic functions before, but have you ever considered quadratic inequalities? A quadratic inequality is any inequality that fits the form:

$y > a x^{2} + b x + c$

The specific inequality symbol used doesn't matter for classification purposes, so we could replace the > above with <,≥, or ≤ and still be working with a quadratic inequality. In this article, we'll explore how to graph quadratic inequalities. Let's get started!

The procedure for graphing quadratic inequalities

Graphing quadratic inequalities begins by graphing the parabola associated with the quadratic equation. The parabola is a solid line when the symbol is ≤ or ≥ because the values on the parabola are included in its range. If we have a < or > symbol, we draw the parabola with a dotted line to indicate that the parabola itself isn't included in the range. Then, we shade the area above or below it depending on the inequality symbol we're working with (greater than or less than).

All of that might sound a little abstract, so let's try a sample problem by graphing $y \leq x^{2} - x - 12$ . The quadratic equation is:

$y = x^{2} - x - 12$

Since a is 1 in $a x^{2} + b x + c$ format, we know the parabola will point upward. Likewise, the c value of -12 indicates that the y-intercept will be (0, -12). We need more points to work with though, so let's try factoring the equation.

Our b value is -1 and our c value is -12, so we need a factor pair with a sum of -1 and a product of -12. If we list out all of the factor pairs for -12, we'll find that the only ones with a sum of -1 are -4 and 3. That means we can factor the equation like this:

$y = (x + 3) (x - 4)$

Setting each of these factors equal to zero gives us x-intercepts at -3 and 4. The vertex of the parabola must be in the middle of these two points, so its x-coordinate will be 0.5. We can plug 0.5 for x in our factored equation to determine the y-coordinate:

$y = (0.5 + 3) (0.5 - 4)$

$y = 3.5 \times (- 3.5)$

$y = - 12.25$

The vertex's coordinates are (0.5, -12.25). With this, both x-intercepts and the y-intercept, we have all of the information we need to sketch the parabola. Remember to use a solid line since our inequality symbol is "less than or equal to:"

The final step is shading. The easiest way to determine whether we should shade the inside or outside of the parabola is to plug in a sample point, with the origin (0, 0) typically leading to the easiest math. Doing that in this case, we get:

$0 \leq 0^{2} + 0 - 12$

$0 \leq - 12$

That's a false number statement, so we shade the part of the parabola that does NOT include (0, 0). Our image should look something like this when we're done:

There are a lot of steps involved, but it shouldn't be too difficult as long as we pay attention to what we're doing.

Sketch the following inequality: $y \geq 2 (x + 1) 2^{2}$

The first step is sketching the parabola, and the easiest way to do that, in this case, is to look at $y = x^{2}$ as a baseline. We're shifting the graph of the base equation one unit to the left and vertically stretching it by a factor of 2, giving us a graph that looks like this:

Since the parabola is included in the range, it should be drawn with a solid line. Next, we have to decide whether we're shading inside or outside of the parabola. The best way to do that is to plug in a random point to see if it satisfies the inequality. Let's use the origin point, (0, 0):

$0 \geq 2 (0 + 1) 2^{2}$

$0 \geq 2$

That's NOT true, so we shade the portion of the graph that doesn't include the origin. Once we shade the inside, our final answer looks like:

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