Hi, and welcome. My name is Anthony Varela, and today we're going to be graphing quadratic inequalities. So first, we're going to talk about how to graph a parabola using an equation, then when talking about inequalities, we'll have to decide to use a dashed line or a solid line based on our inequality symbol, and then how to shade a solution region to represent solutions to our quadratic inequality.
So let's start off by graphing a parabola. So here we're given the equation y equals 1/2 x squared minus x minus 4. Now, the key to graphing a parabola, especially when were given the equation in standard form, is to locate that axis of symmetry and the vertex to the parabola using this formula, x equals negative b over 2a. And we draw our values for a and b from the x squared coefficient, this would be a, and the x term coefficient is b.
So finding the axis of symmetry, we're going to solve for x using negative b over 2a. So we can see that b is negative 1, so negative b would be positive 1. And then a is 1/2, so 2a would be a positive 1 as well. So we have x equals negative b over 2a corresponds to x equals positive 1 over positive 1. So x equals 1.
So on our graph then for our parabola, we know that our axis of symmetry is the line x equals 1, and this is going to help us locate our vertex because the vertex of the parabola is on the axis of symmetry and will also allow us to plot points on one side of the axis of symmetry and then this reflect across that line to plot another point.
So let's find first the y-coordinate of our vertex. So we're going to plug in 1 into our equation. So when x equals 1, 1/2 x squared equals 1/2 minus x is minus 1, and then we have minus 4. So our y value is negative 4.5. So at x equals 1 and y equals negative 4.5, we have our vertex to our parabola.
So let's pick some other x values. When x equals 2, 1/2 x squared equals 2 minus x equals minus 2 and then we have minus 4. So we can plot then the point 2, negative 4, and we can reflect this across our axis of symmetry.
Let's choose another x value. When x equals 4, 1/2 x squared equals 8 minus x is minus 4, and then we subtract 4. So we can plot the point then for 0 on our graph. We could also reflect this across our axis of symmetry, and now we have enough points where we can draw our parabola shape. So there is the graph for y equals 1/2 x squared minus x minus 4.
Now, when graphing inequalities, we have to think about using a dashed line or a solid line, depending on the inequality symbol. So we call this our boundary line. The parabola is the boundary line to our inequality. And when we have strict inequality symbols, we're going to be using a dashed line, so here are strict inequality symbols. If we have non-strict inequality symbols, we're going to be using a solid line to graph our boundary line.
And then we're going to be shading in a solution region, and this is going to represent all points, xy, that satisfy our inequality. So let's go ahead and graph this inequality. y is less than or equal to 2x squared minus 8x plus 5. So once again, I'm going to first identify the axis of symmetry, find the vertex, and then we'll plot some points so that will help us then draw our boundary line.
So using x equals negative b over 2a, well, we can see that b is negative 8, so negative b would be positive 8. And then a is 2, so 2a would be 4, so you have x equals 8 over 4. So we know then that the axis of symmetry is at x equals 2. So there is our axis of symmetry. We're going to plug in that x value of 2 into our inequality, solve for y. That's going to give us the y-coordinate of our vertex.
So when x equals 2, 2x squared is 8 minus 8x is minus 16, and then we have plus 5. So we can plot the point then 2, negative 3. That represents the vertex of this parabola, which represents the boundary line toward inequality. So now when x equals 3, we can say then that 2x squared is 18 minus 8x is minus 24, and we have plus 5. So we can plot the point 3, negative 1 and reflect it about that axis of symmetry.
When x equals 4, 2x squared is 32 minus 8x is minus 32, and we have plus 5. So we can plot the point 4, 5 on our coordinate plane and reflect it about our axis of symmetry.
So now we're going to draw our boundary line, and because our inequality symbol is a non-strict inequality symbol, we're going to be using a solid line to graph our boundary line. So here so far is our inequality without the solution region shaded. So to shade part of the solution region, one thing we could do is we could look at our inequality symbol, and this would tell us to shade everything below the boundary line-- that would be part of our solution region-- but it can be sometimes a little bit confusing to figure out exactly what is below the line when we have a U-shaped parabola here.
So another way to find the solution region is to pick a test point, and I love choosing the origin because that's 0, 0, and it's really easy to plug in 0 for x and do some calculations. And so we're going to be substituting 0 in for x and 0 in for y and see if we get a true statement. So this would just be 0 because we have 0 squared times 2. This is also 0. So now we have these statements that 0 is less than or equal to 5, and this is a true statement.
So that means that this test points, our origin 0, 0, fits within our solution region, and we can shade that entire region. So this would be the solution region to our quadratic inequality.
So let's review graphing quadratic inequalities. Well, we went to first graph the parabola, so this would be y equals, so ignoring the fact that it's an inequality for now. We want to find the axis of symmetry and the vertex. That's going to help us plot points so we can eventually graph that parabola.
Now, in inequality, we graph a boundary line, and so we're going to be using either dashed lines or solid lines depending on the inequality symbol. And then we need to identify a solution region. So these are all points, x and y, that satisfy the inequality. A great method is to choose the origin as the test point and see what happens when you substitute x and y, or 0, 0, in for x and y, and if you get a true statement or not.
So thanks for watching this tutorial on graphing quadratic inequalities. Hope to see you next time.