Find the vertex, focus, and directrix of the parabola with the gi...

Question

Find the vertex, focus, and directrix of the parabola with the given equation. Then graph the parabola. (y-2)^2 = -16x

Accepted Answer

Hello. Today we're going to be using the given equation to identify the graph of the parabola. So what we are given is y plus one squared is equal to negative for X. So this is given to us in the standard form of a parabola that's not located at the origin. And the standard form is going to be y minus k squared is equal to four p times x minus H. So before we start anything, let's go ahead and identify the center of the parabola. And we can do that by looking at the X and y quantities. See the center is given to us in the form of h comma K. If we take a look at the X Quantity X can be rewritten as X zero, which is in the standard form of X -H. So in this case here H is going to equal to zero. And if we take a look at the y quantity we have Y plus one, this is not in the standard form of y minus k. But we can rewrite this to give us why minus negative one. And this is going to be where K is equal to negative one. So since we've identified H and K, putting this into the center is going to give us a center of zero comma negative one. This is going to be the vertex of the parabola. And one more thing to identify since the y term is squared in this case this is the equation of the parabola that's going to open up to the left of the x axis. So let's go ahead and identify where the focus and direct tricks is going to be located at. We need to first find the distance of the focus from the vertex and that is given to us by the focal constant four p. In the given equation four P is equal to negative four. So if we want to identify the distance of the focus, we're going to set four P equal to negative four, Dividing both sides by four is going to give us a value of P is equal to negative one. So that means that the focus is going to be located one unit to the left of the vertex. So we're going to subtract one from the X value of the vertex that's going to give us zero minus one, comma zero. Which is going to simplify to negative one comma one and that's going to be the location of the focus. Now the direct tricks is going to be equal distant from the vertex. So if the focus is negative one unit to the left of the vertex, the directory is going to be located to the right one unit. And so if we take a look at the X value of the vertex, X is currently equal to zero. But if the directory is located one unit to the right, we're going to add one to this value and that means that the graph of the directory IX Is going to be X is equal to one. So now that we've identified the direct tricks the vertex and the focus, let's go ahead and choose the option that accurately represents all these values. And that option is going to be a see here. The Vertex is located at zero Common -1. The direct tricks is given to us as X is equal to one and the focus is located one unit to the left, which is at negative one comma one, which matches all the descriptions that we've come up with. So that means that the answer to this problem is going to be a So I hope this video helps you in understanding how to identify the graph of a parabola. And I'll go ahead and see you all in the next video.

Find the vertex, focus, and directrix of the parabola with the given equation. Then graph the parabola. (y-2)^2 = -16x

Key Concepts

Parabola Definition

Vertex, Focus, and Directrix

Graphing a Parabola

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