How to write parabola equations...?

Absolutely. First let's write out all the different equations associated with parabolas, then we can choose which we need based on the information given.

Parabola's opening up or down (U or ∩):

y = A(x - h)² + k

A is the constant which determines the shape of the parabola. If A is positive the parabola opens up, if it is negative it opens down. The point (h,k) is the vertex.

Parabolas opening left or right (⊂ or ⊃)

x = B(y - k)² + h

If B is negative it opens left, positive it opens right. It should be noted that these parabolas are not functions as they don't pass the vertical line test.

Alright, with all that let's get the information we know. The vertex is at (5,-3), so we have h = 5, k = -3. This was the easy part, it gets a bit harder from here. Okay, so the focus is inside parabola. So because the focus is 3 units to the right of the vertex we're going to use the second equation and B will be positive. To find B we have to use the definition of a parabola, that is that it is the set of points that is equidistant from a point (the focus) and a line (the directrix). Because we have the focus and vertex, which we know are 3 units apart, the directrix is easy to find. It's 3 units the other way, and because the parabola is opening to the right, it is a vertical line. The directrix is x = 2. Now, to find B, we need another point on the parabola, any point other than the vertex will do. So we're going to find the point on the parabola directly above the focus, this is the easiest to find (or the one below the focus). So we know the x coordinate of this point is 8, and the distance to the directrix is 6. So we need the distance to the focus to be 6 as well, but since they're on the same vertical line this boils down the distance between the y-coordinates. So our point is (8,3). Plug this point and the vertex into the formula for the parabola to solve for B.

8 = B(3 - (-3))² + 5

3 = B6²

3/36 = 1/12 = B

So our parabola is

x = (1/12)(y + 3)² + 5.

It should be noted that there are formulas to find A and B, but they're based on the same exact process I walked you through, just in general. Try doing this process in general if you're interested in those formulae.

I hope this has helped. Good luck, and don't forget pictures are your friend. Have a nice day!

observe that query asks you to write down AN equation, no longer THE equation of parabola One such equation is y = (x - a million) (x - 12) = x^2 - 13x + 12 the final equation of parabola with x-intercepts a million and 12 is y = a (x - a million) (x - 12)