In Exercises 17–30, find the standard form of the equation of eac...

Question

In Exercises 17–30, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (0, 15); Directrix: y = - 15

Accepted Answer

Hello everyone in this video. We're going to be looking at this practice problem where we want to write the equation of a problem and standard form given the conditions that it has a focus at 012 and a direct tricks at why equals negative 12. So I'm going to go ahead and start with a picture to sort of visualize what this problem looks like. So I'm going to label this my Y axis and my X axis and I don't want it to be super precise picture, I just kind of want to use it to visualize. So I'm going to label My y axis, I'm going to say this is negative and above the X axis this is positive 12. I'm just gonna have those two labeled. I'm gonna go ahead and start graphing what is given to me. So first I have a focus at 0 12. So I'm gonna go ahead and draw a dot there and I have a direct tricks at Y. Equals negative 12. So I'm gonna do a dotted line across where Y equals negative 12 and only for that Y equals negative 12. And I know that that's my direct tricks. So recall that problem has opened up towards the focus and away from the directorate. So I know that my problem opens up upwards. I'm not exactly sure where yet but I know that it opens up upwards. So from here you can see that my access of cemetery or where the vertex is at is along the Y axis. You can also sort of derive this because the direct tricks is given in terms of why if the direct tricks is given in terms of why then I know that my standard form is going to have V. X. On the left hand side. So I'll have x minus H squared is equal to four p times y minus K. And in this standard equation you can see that there's three sort of variables that we need to determine. We have H. P. And K. And one of the ways that we can knock two of them out is by finding the vertex because the vertex is actually H. Kay and in this case the root tex is the halfway point between the focus and the direct trick. So if we look at this graph, the vertex falls halfway between these two points. And in this case this graph makes it easy to visualize because We have our focus at positive 12 and our direct tricks at -12. So naturally the halfway point is at the zero mark. We can also determine this mathematically. So for our vertex you can see that it falls in the same X. Value as the focus. So we already know that H is zero and K would be 12 plus negative divided by two. And here you can see that this becomes zero and the numerator Become zero divided by two. So my coordinates become 00. So the same value that we visualized on the graph is what we determined mathematically. And more importantly we found H and K. So I'm going to go ahead and plug those into my equation. So I have a church or x minus H. Which is zero squared is equal to four P times y minus K, which is zero. So my equation becomes x squared is equal to four p times y. So now I only need to determine P. And P is the distance from my vertex to my focus. And in this case the distance between my Focus and Vertex is 12. You can see that from the graph here. P is 12 because I start at zero and go to 12. We can also determine that mathematically. We start with the Why value of the focus which is 12 to the vertex is Y. Value which is zero. I got 12. So P is equal to 12. And I can plug that into my equation. So I have x squared is equal to four times P, which is 12 times Y. And I do that multiplication, I have x squared is equal to four times 12, which is 48. Why? So this becomes my final answer and let me just go back to this graph really quick because I don't I never drew in my problem but I know that it opens upward and I know that my vertex is at 00. So I can go ahead and draw it in and that is what the problem would look like. Um and notice how we already knew that it was going to open upwards, because we were given the focus and the direct tricks, so we knew that it was going to be opening up towards the focus. But this X squared is equal to 48 Y becomes our final equation, and you can see that that aligns with the answer choice D. So hopefully this video helped and see you next time.

Struggling with College Algebra?

In Exercises 17–30, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (0, 15); Directrix: y = - 15

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