Question

You are given the coordinates of the vertex (1,6) and of a point (-8,-9) on a parabola. find the equation of the parabola.

Answer 1

We know this is an upside down parabola because the vertex is higher than the point. So it will have a negative leading coefficient or "a" value. We have this then as our standard form:
`y=a(x-h) ^(2)+k` where h and k are the coordinates of the vertex and x and y are the coordinates of the point. We will fill in our standard equation with those values and solve for a, then rewrite. Here's what we have so far:
`-9=a(-8-1) ^(2) +6`. Simplifying that will give us -9=a(81)+6. Subtract 6 from both sides and we solve to find that a=-5/27. See, that's how we know it's upside down. Our a value is solved as a negative. Now, put that value of a back in along with the vertex to get a final equation of
`y=- (5)/(27)(x-1) ^(2)+6`