In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola.

x^2 = - 16y

Question

In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola.

x^2 = - 16y

Accepted Answer

Hey, everyone here we are asked to solve for the focus and theatrics as well as graph. The given equation of a Parabola where we have X squared is equal to negative 24 Y. Here we have four answer choice option ABC and D. Each of which display a graph for our given equation of a Parabola. And they include the di tris in the graph as well. And each solution states the focus and the dix. So to begin solving this problem here again, we have the given equation X squared is equal to negative Y. And now we can rewrite this equation of a Parabola in its proper form where we have the quantity of X minus H squared is equal to four times a times the quantity of, of Y minus K. So with this formula, the standard form of a Parabola, we can compare this formula with our given equation for a parabola and identify our focus and vertex. So we can first identify our value for A. So in the formula we see that A is has a coefficient of four. So in our equation here we have negative 24. Well, negative 24 divided by four gives us negative six. So we see here that A is going to be negative six. And so now we can identify the vertex first. And if we recall the vertex is the point H K, so in our case, the point H K is just going to be at the origin or at zero and zero. And now we can identify the focus. Well, the focus is the point H and K plus A. So now comparing to our given equation and formula here, our focus is going to be zero and zero minus six. Simplifying, we see the focus is at zero and negative six. And now we need to note that the Di Rics and the focus lie at equal distances from the vertex of the parabola. And the equation of the dire can be calculated based on the equation of the Parabola. So since the vertex is at the origin 00 and the focus is at A, we can find the equation of the dix at Y is equal to negative A. Well, at Y is equal to negative A A, we have Y is equal to negative quantity of negative six which gives us theatrics is at Y is equal to positive six. So now we have identified the dia at Y is equal to six. We also have our focus at zero and negative six. And we also have our vertex identified. So we can go ahead and graph our parabola as well as the di and so in doing this, we have our graph of our parabola which is a downward facing parabola. With again the vertex at the origin, we see the focus is at zero and negative six. And we see this dash line is our, so with this graph, as well as our components identified, we see here that we are left with answer choice C where again, we have our graph of our parabola and we also have our focus at zero and negative six and ours at Y is equal to six. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola. x^2 = - 16y

Verified Solution

Key Concepts

Parabola Definition

Focus and Directrix

Graphing Parabolas