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.Vertex: (2, - 3); Focus: (2, - 5)

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 the problem in standard form with the conditions that the vertex is at 1 - and the focus is at 1 -9. So I'm going to start this problem by drawing a picture to help me visualize what is happening. So I'm gonna do the Y axis and then I'm going to draw the X axis and I'm not going to make this picture very precise. I just want to make sure that I can sort of visualize my proble. So if I draw my vertex I'm just gonna draw one and then I'll go ahead and number my why access so go 123 and then negative one negative two negative three -4 -5 -6 -7. And for my x. axis I'll go go negative one negative two negative three negative four. Okay so now I have my labeled X. And Y axis. Now I'm going to draw in my or drawing my points. So this first point I have the vertex at one and negative four. So there's my four attacks and then my focus is at one and negative nine. Gonna go ahead and add some ticks. So negative eight negative nine. So still one negative nine. It's not how both of these points. And recall that the Parabola opens up towards the focus which means that I know that my parabola opens down in some way. So now I've drawn sort of guess of what the problem would look like since I know that my vertex is at one in negative four. And it opens up towards the focus. So from here we can start to determine the standard form equation for this problem. So because it opens up or down you can see that the access access of cemetery is sort of on the y axis. So it's going to follow the form of x minus h squared is equal to four P. Why minus K. And in the standard equation you can see that we have three variables that we need to find. We need to find a church P and K. Well remember that the vertex is actually equal to H. Okay, which means that we've already found two of these variables. Since we're given the vertex the vertex is one -4. So I'm going to go ahead and plug those in. So my standard form equation becomes X -H, which is equal to one Squared is equal to four p times y minus K. And in this case K is equal to negative four. So I have two negatives. So write that as plus four. So notice I plugged in the values from the vertex into my equation. Now I only have one variable that I need to solve for which is P. And in this case recall that P is the distance from the vertex to the focus. So this length here Is P and we can determine that because we drew it. So if I count how many spaces it takes up 12345. So I can say P is equal to five. But if you had it drawn it can also see that we can determine it from the y. Values of the focus and the vertex. So if we find P going from the focus to the vertex I need to worry about these. Why values where I have negative nine minus negative four. That becomes negative nine plus four or negative five. So now I know that P is equal to -5. So the distance here from the focus to the vertex is negative five. And I can also plug that into my standard form equation. So I have x minus one squared is equal to four times P. Which is negative five times Y plus four. So if I do that multiplication I'm left with X minus one squared is equal to four times negative five is negative Times y Plus four. And this would become this is my final answer for the equation. And this aligns with answer choice A and notice how we have the negative because our problem opens downwards. So always make sure to check that the equation makes sense with what you're visualizing the problem and in this case we drew it so it makes sense. But 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.Vertex: (2, - 3); Focus: (2, - 5)

Watch next

Parabolas practice set