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Graphing Quadratic Inequalities

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Today we're going to talk about graphing quadratic inequalities. So we'll start by reviewing how to graph a quadratic equation, and then we'll do an example graphing a quadratic inequality.

So let's start by reviewing how to graph a quadratic equation. I've got the equation y is equal to negative x squared minus 4x minus 3. So I'm going to start graphing it by using the formula to find the x-coordinate of the vertex and then substituting that value to find the y-coordinate of the vertex. So the formula for finding the x-coordinate is x is equal to negative b over 2 times a. I know that my a value is going to be negative 1 and my b value will be negative 4. So substituting that into my equation, I'll have a negative negative 4, which will give me positive 4.

And 2 times negative 1-- simplifying this, that gives me 4 over negative 2, which is negative 2. So I know the x-coordinate of my vertex is negative 2. Substituting negative 2 back into my equation, I can find the value of y for the vertex. So I'll have negative negative 2 squared minus 4 times negative 2 minus 3.

Simplifying this, I'm going to first square negative 2, which will give me positive 4. And then I'll bring down the negative that's in front. Then I've got minus 4 times negative 2, which is negative 8 minus 3. Simplifying this-- negative 4 minus negative 8 is going to give me a positive 4. And 4 minus 3 will give me a y value of 1. So I know that my vertex is at the point negative 2, 1.

So I'll start by plotting a point at negative 2, 1 for my vertex. Now, I notice that in my equation, the coefficient in front of my x squared term is negative. You can think of this is a negative 1. So I know that my parabola is going to open downwards, facing downwards, and so that this vertex will be a maximum point. So I'm going to find two other points on one side of my vertex, and then I can use my axis of symmetry as a reflection line to reflect those points on the other side. And then I can get a pretty good idea of the shape of my parabola.

So I'm going to go ahead and use the two x values of negative 1 and 0. So I'm going to start by substituting negative 1 for x into my equation. So I'll have y equals negative negative 1 squared, minus 4 times negative 1 minus 3. Simplifying this, I'll have positive 1, with this negative in front, minus a negative 4 minus 3. Negative 1 minus negative 4 will give me positive 3, and 3 minus 3 will give me 0. So I know that when x is negative 1, y will be 0. So I can plot a point there.

And then using an x value of zero, substituting that in for x in my equation, I'll have 0 squared is just 0 minus 4 times 0 is 0 minus 3. So this will give me a y value of negative 3 when x is 0. So now I can plot these three points, and I see that my parabola looks a little bit like this. And again, I can use my axis of symmetry, which is this vertical imaginary line through the vertex, and I can plot a couple more points on the other side. So this point will also be one away from the axis of symmetry, and the point over here will be two away. And I can find a rough graph of the parabola, which is represented by this equation.

So here's an example of graphing a quadratic inequality. When we're graphing a quadratic inequality such as y is less than x squared minus 2x plus 1, we graph it in the way that we graph a quadratic equation. And then, similar to the way that we graph a linear inequality, we use either a solid line if the inequality symbol is less than or equal to or greater than or equal to, or we'll use a dotted line if the inequality symbol is strictly less than or strictly greater than.

And then we'll shade either above or below the line of the parabola. We'll shade below if the inequality symbol is less than or equal to or strictly less than, and we'll shade above the line if the inequality symbol is greater than or equal to or strictly greater than. So let's start by seeing what the shape of a parabola would be if we were to graph the equation y equals x squared minus 2x plus 1.

So to do that, we'll start by finding the x-coordinate of the vertex using the formula x is equal to negative b over 2a. I know that my value for b is in front of my x term, so that will be negative 2. So this is a negative negative 2, which will become positive 2. And my a value is positive 1. So 2 over 2 gives me an x-coordinate of my vertex of 1.

I can find the y-coordinate of my vertex by substituting this value of 1 in for my x's to find y. And again, I'll replace the inequality symbol with an equal sign, so this will become 1 squared minus 2 times 1 plus 1. 1 squared is 1. 2 times 1 is 2. 1 minus 2 will give me negative 1. Plus 1 will give me a y value of 0 for my vertex. So I can start by plotting the point 1, 0, and I see that my vertex will be here.

Now I can find two other points on my parabola by picking two x values and substituting them in to the equation to find the corresponding y values. So I'm going to start with an x value of 0. So I'll substitute 0 in for x, and I'll have 0 squared minus 2 times 0 plus 1. 0 squared is 0. 2 times 0 is 0. So I will have a y value of 1 when x is equal to 0. So plotting that point.

Then I'll use the x value of negative 1. Substituting that into my equation-- negative 1 squared is positive 1. 2 times negative 1 will give me negative 2. 1 minus negative 2 will give me 3, and 3 plus 1 will give me 4. So I found that my y value is 4 when x is negative 1. So I'll plot the point negative 1, 4, which will be here.

So now I can find the two other points that are on the other side of my axis of symmetry, which goes through the vertex. So I'll see that another point will be one away from here, and another point will be one, two away. So, one, two-- so here. And now I can connect these points. I need to see if I should use a solid line or if I need to use a dotted line. So again, looking at my inequality symbol, I see that it's strictly less than. So instead of using a solid line, I need to actually connect these points using a dotted line or a dashed line. So it will look something like this.

And now, because I see that my inequality symbol is less than, I know that I'm going to be shading below my parabola. But I can also pick a point to test to make sure that that is the solution region. So I'm going to go ahead and pick the point 0, 0, when x is 0 and y is 0, and see if by substituting 0 in for x and y into my inequality I'll yield a true statement.

So if I substitute 0 in for y and 0 in for my x's, simplifying this side, I'll have 0 minus 2 times 0 is 0, plus 1. So this will give me 1. And I see that 1 is greater than 0. So that means that because the values of 0 for x and y yield a true statement, I know that the point 0, 0 is within my solution region for this inequality. So I'm going to go ahead and shade everywhere below my parabola.

So let's go over our key points from today. To graph a quadratic inequality, we graph the parabola as we do with equations, but use either a dotted or a solid line, depending on the inequality. Inequality symbols less than and greater than indicate using dotted or dashed lines, and inequality symbols less than or equal to or greater than or equal to indicate using solid lines. The solution to the inequality can be represented by shading a portion of the plane. If a test point yields a true inequality statement, then the region the point lies in is shaded. If a test point yields a false inequality statement, then the other portion of the plane is shaded as the solution region.

So I hope that these key points and examples helped you understand a little bit more about graphing quadratic inequalities. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.