Lesson 5-7: Graphs of Quadratic Inequalities

We’re almost there! This is the last lesson on quadratics. Woot! 

I don’t know if you’ve noticed but the path we’ve taken through quadratic land has been very similar to the path we took through linear equation land. There is one remaining concept we need to cover for quadratics that we played with in linear land. It is the concept of inequalities tied to quadratics…quadratic inequalities.

Quadratic inequalities

I’m sure you can guess what a quadratic inequality is. They come in four flavors:

  1. Greater than >
  2. Greater than or equal ≥
  3. Less than <
  4. Less than or equal ≤

Why don’t you try to write a version of the standard form quadratic using each of the inequalities?

  1. Greater than: y >ax2 + bx + c
  2. Greater than or equal: y≥ax2 + bx + c
  3. Less than: yax2+ bx + c
  4. Less than or equal: y ≤ ax2 + bx + c

Graphs of quadratic inequalities

Don’t panic! This is easier than it sounds! If you can graph a regular quadratic equation and remember how we handled linear inequalities, you will be just fine!

The process is almost identical for a quadratic equation! The only difference is that instead of graphing one point, pick two: one inside the U of the parabola and one outside. Why? Because it will help you catch errors in graphing better!

Here is the process to graph a quadratic inequality:

  1. Graph the inequality as an equality (y = ax2 + bx + c)
  2. Solid if the inequality is ≤ or ≥
  3. Dashed if the inequality is < or >
  4. Test a point inside the U and one outside the U…only one can work!
  5. Shade the side that contains the point that is a solution for the inequality.

Determining if a point is a solution for a quadratic inequality

To determine if a point is a solution for a quadratic inequality, just plug in the x-coordinate for x and the y-coordinate for y and simplify. If you come up with a true statement (like 3 < 4) then the point is a solution for that inequality. Let’s try a few:

Why are we practicing this? How do we decide which side of the parabola U to shade? That’s right! By testing points; in other words, by seeing if the points are solutions of the inequality.

Sketching the graph of a quadratic inequality

Let’s try one out here: graph:

x / y
0 / -3
1 / 3
3/2 / 15/4 = 3.75
2 / 3
3 / -3

Since the inequality is < the parabola will be dashed.

Pick and test a point inside the U and one outside. Pick the easiest possible!

Inside: (1, 0) Outside: (0, 0)

So we shade the area inside the parabola.

Sketching the graph of a quadratic inequality

Now let’s consider another quadratic inequality at the same time as the one above. We need to graph both and then only shade the region that both cover. This is called the intersection of the two graphs.

Graphand the above inequality:

x / y
0 / 4
1 / -2
2 / -4
3 / -2
4 / 4

Since the inequality is > the parabola is dashed.

Pick and test a point inside the U and one outside.

We can use the same two we used for the first inequality:

Inside: (1, 0) Outside: (0, 0)

So we shade the area inside the 2nd parabola.

The intersection of the two graphs is the weird oblong shape in the middle!

Page 1 of 4