Hi, and welcome. My name is Anthony Varela. And today, we're going to graph parabolas.
So we're going to see an example where our equation is given to us in standard form. We're going to see a second example where our equation is in vertex form. But in both of our examples, we're really going to be using the vertex and the axis of symmetry to help us plot points.
So our first example is going to be graphing in standard form. So standard form is y equals ax squared plus bx plus c. So let's graph y equals negative 2x plus 8x minus 6.
Now one of the easiest points to graph when we have our equation in standard form is the y-intercept. Now the y-intercept has an x-coordinate of 0 as do all y-intercepts. Now when x equals 0, our x squared term is 0. Our x term is 0. So really the y-coordinate of the y-intercept is defined then by our c values-- so in this case, negative 6.
So we already know then that the point 0 negative 6 is on our parabola, but really what we want to do is find the vertex of this parabola. That's going to help us plot points, because the vertex represents either that minimum or maximum point. It also lies on the axis of symmetry. So we can plot one point. And then just reflect across that line to plot another point.
So how do we find the vertex in standard form? Well, we can describe generally the vertex is having coordinates x and y. In standard form, we can find the x-coordinate of the vertex by using x equals negative b over 2a. So we get a and b from these coefficients here.
So we see then that negative b would be negative 8, and 2a would be negative 2 times 2. So negative 8 divided by negative 4, I can just say that's positive 8 over positive 4. So x value is 2.
So we know right now then that our vertex has an x-coordinate of 2. To fine y, we just plug 2 in for x and solve for y. So when x equals 2, negative 2x squared would be negative 8 plus 8x would be plus 16.
And then we have our minus 6. So y equals 2. So we know that our vertex then is at the point 2.2. And I've also drawn in that axis of symmetry. So now what we can do is just choose a couple of x values that are on one side of the axis of symmetry.
Find the corresponding y value and then reflect that point. So when x equals 3, we have negative 2x squared equals negative 18 plus 8x is 24 and then minus 6. So when x equals 3, y equals 0.
So we can plot the point 3, 0. We know that's going to be in our parabola. And we can just then reflect across our line of reflection. So we know that this is also a point on our parabola.
So now let's pick the point x equals 4. So when x equals 4, negative 2x squared is negative 32. 8x is positive 32. And then we have our minus 6.
So y equals negative 6. So we know then the point 4 negative 6 is on our parabola. And we kind of reflect that across this line of symmetry. We already have it on there This is our y-intercept.
So now we have a couple of points on our graph. This is what our parabola looks like. Let's go through an example of graphing in vertex form. So vertex form is y equals a times x minus h quantity squared plus k.
So here is our equation. y equals 1/2 times x minus 2 quantity squared, minus 5. Now what we want to do right away is find the coordinates of the vertex, which is pretty easy in vertex form. So our vertex can be defined by h comma k.
So h corresponds to the x-coordinate. And k corresponds to the y-coordinate. So we can see then, in this case, h equals 2, and k equals negative 5. So I have found 2 negative 5. There is our vertex.
And now we can use the same strategy as before, picking a couple of x values, finding their corresponding y values, and reflecting, because we know the vertex is on our line of reflection. So when x equals 4, then we have y equals 0.5 times 4, because 4 minus 2 squared is 4. And we can take away 5. So y equals negative 3.
So we know then that the point 4 negative 3 is on our parabola. And we can reflect. So we know that the point 0 negative 3 is also on our parabola.
And if you're having trouble reflecting, just take a look. This was two units away from that imaginary axis of symmetry. So then I'll go 2 units away in the other direction.
Let's pick another x value. X equals 6. When x equals 6, y is 0.5 times 16, because 6 minus 2 squared is 16. We take away 5. So we see that y equals positive 3.
So we know that the 0.63 is also on our parabola. And remember, this is where our line is symmetry was. So we can count 1, 2, 3, 4 units. So then we'd go 1, 2, 3, 4 units here. So that's going to be a point on our parabola.
Now that we have a couple of points, we can go ahead and connect the dots and there is our parabola for this equation. So let's review graphing parabolas. Well, if we have our equation written in standard form, you can find the y-intercept by just looking at your c value. But what you really want to do is find your vertex.
So the x-coordinate of the vertex is negative b over 2a, and then just find it's associated y value. And you can use then that line of reflection to reflect a couple of other points. If you have your equation in vertex form, finding the vertex is pretty easy. Just look at h and k.
The h value is the x-coordinate of the vertex. And k is the y value of the vertex. And once again, once you have that vertex on your graph, you can find a couple of other points to plot and just reflect over that axis of symmetry. So thanks for watching this tutorial on graphing parabolas. Hope to see you next time.
