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

x^2 = 12y

Question

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

x^2 = 12y

Accepted Answer

Hey, everyone for the following equation of a Parabola, we are asked to solve for the focus and the direct tricks. And then graph here, the given equation is X squared is equal to Y. Here we have four answer choice options. Each of which display a graph representing our Parabola as well as the direct tricks. And each solution states the focus and the direct tricks. So to begin solving this problem first recalling that our given equation here is X squared is equal to 24 Y. Well, we see that this given equation follows the form where we have the quantity of X minus H squared is equal to four A times the quantity of Y minus K. So with this formula, we want to first identify our value for a by comparing the formula to our given equation. So we see the coefficient of Y in our equation is 24. So we can equate 24 with four a from our formula. And now dividing both sides by four, we see that our value for A, in our case is going to be positive six. So now with a identified, we can move on and identify our vertex. Well, if we recall the vertex is at the point H K. And so using our given equation, we see that the vertex, in our case is going to be at the origin or at zero and zero. Now we can also identify the focus. So the focus is going to be at the point H for the X coordinate and then at the point K plus A for the Y coordinate. So substituting in our information that we have, we see that the focus is going to be at zero and zero plus six. And so simplifying the focus point is going to be the 60.0. and six. So now we also to note that the direct tricks and the focus lie at each distances from the vertex of the parabola. And the equation of the direct tricks can be calculated based on the equation of the Parabola. So since the vertex is at zero and zero and the focus is at a, then we know the equation of the direct tricks is going to be at Y is equal to negative A and negative A is just the negative quantity of positive six. So we see here that our direct Trix is going to be at Y is equal to negative six. So now with our direct tris identified as well as the vertex and our focus, we can now graph our parabola and the direct tricks. And so in doing so we see we now have our parabola which is an upward facing parabola with its vertex at the origin. The focus is again at the 0.6. And the, we have this dash horizontal line at Y is equal to negative six to represent our direct tricks. So now with this graph and our identified pieces of information, we can now compare our solution here with our answer choice options. And we see that we are left here with answer choice D where again, we have a focus at zero and six and the direct tricks is Y is equal to negative 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 = 12y

Verified Solution

Key Concepts

Parabola Definition

Focus and Directrix

Standard Form of a Parabola