This week in PreCalc 11, we learned about graphing quadratic inequalities with two variables. Inequalities state whether two values are equal, not equal, greater than or less than the other. The point of the inequality is to make it true, If the inequality states that then it’s true because 7 is greater than 6, but if it states , then it’s not true because 7 isn’t less than 6.

Inequalities can be expressed by quadratic equations. For example . We are supposed to graph this quadratic equation in order to find possible solutions that make the statement true.

Step 1: Graph

To start this process, we have to graph the quadratic. We can start graphing this quadratic by one of two ways. One way is to convert this inequality from general form to vertex form. We can do this by the completing the square method. This form tells us where the parabola is located and what the stretch pattern is so we know the pattern to go up by. Another way we can graph this is to factor it. To factor we have to find two numbers that when multiplied equal -8 and when added equal -2:

In this case the numbers were -4 and 2.

Now we know what the zeros of the equation is so that means the opposites are the x-intercepts. The opposite of -4 and 2 are 4 and -2 which means that those are the x-intercepts of the graph. Since we know where the x-intercepts are, we can find out the vertex. We can do this by adding the x intercepts together and dividing them by 2.

Now we know that 1 is the x value of the vertex. To find the y value, we can put the x value into the original equation to solve for y.

Now we have the y-value which means we know the vertex

We also know that the stretch value, the value that determines the pattern the parabola follows, is 1 because there is no number attached to the in the inequality. That means the parabola follows the pattern of up 1 over 1, up 3 over 1, up 5 over 1, and so on.

The graph should look like the following:

Step 2: Determine the Solution

Now that we know what the parabola looks like, we have to figure out which coordinates on the graph make the inequality statement true. The points that make the statement true are either going to be all the ones inside the parabola or all the ones outside the parabola. To figure this out we can start by picking one point of the graph and dropping the x value and the y value in the original equation. If, when the inequality is solved, the statement is true, then all the points, where its located (i.e. outside the parabola) then all the points outside the parabola are true, so we would shade that area. If it was located inside the parabola then we would shade the inside of the parabola.

When the coordinate is placed in the original inequality, a true statement came out. Zero is greater than negative eight, so that means, since , is inside the parabola, then all of inside the parabola will be shaded.

All the coordinates inside the parabola make the original inequality true.