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. (x-4)^2 = 4(y+1)

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 X plus two squared equal to four times y minus two. Now this is the standard form of the equation of a parabola not located at the origin. And the standard form is given to us as x minus h squared is equal to four P times y minus k. Now, one thing to note here because the h quantity is squared, this is going to be a parabola that either opens up to the top or bottom of the white axis. The leading coefficient in our given equation is positive. So this is going to be a parabola that opens up positively towards the white axis. Now what we need to do is go ahead and identify the vertex which is considered to be the center of the parabola. And since the center is not the origin the vertex is going to be given to us in the form of h comma K. In order to get our H and K values. We need to take a look at the X and Y quantities. So the x quantity is given to us as X plus two but this is not in the standard form of x minus H. So we can rewrite this quantity to be x minus negative two. Which is in the standard form of x minus H. And this is where H is equal to negative two. Next let's get our k value and that's going to be from the y quantity. Why minus two. Now why minus two is in standard form of y minus K and this is going to be where K is equal to two. So the vertex is going to be given to us at negative two comma two. So that's the location of the vertex. Now let's go ahead and identify the focus and we can identify the distance of the focus by taking a look at our focal constant which is four P. In our given equation four P is equal to four. So in order to figure out the distance of the focus from the vertex, we're going to set four P equal to four. And if we solve for P by dividing both sides by four we're going to get P is equal to one. What this means is the focus is going to be located one unit above the vertex. So with that being said we're going to add one to the y value of the vertex and that's going to give us negative two comma two plus one. And that's going to give us negative two comma three. And that's going to be the location of the focus. Now let's identify the equation of the directory. X. The equation of the direct tricks is going to be represented by horizontal line which is going to be y equal to some constant. Now, since the focus is located one unit above the vertex, that means the direct tricks is going to be located one unit below the vertex. So the vertex the y value the vertex is currently equal to two. But if we want to find the value that's below two, we're just going to subtract one and that's going to give us the equation of the direct tricks, which is why it's equal to one. So now that we have the vertex focus and direct tricks, we just need to go ahead and select the option that accurately represents all the information we currently found. And that option is going to be D. C. D. Has the vertex and negative two comma two, which is located here. The focus is going to be one unit above the vertex At -2 03. And the direct tricks is going to be one unit below the vertex, which is represented by why is equal to one. So since D has all the correct information, the answer to this problem is going to be D. 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. (x-4)^2 = 4(y+1)

Key Concepts

Parabola Definition

Vertex of a Parabola

Focus and Directrix

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