Question

What is the equation of the parabola that has a vertex at `(-4, 2)` and passes through point `(-7,-34)`?

Answer 1

To solve this you need to use the vertex form of the equation of a parabola which is `y=a(x-h)^2+k`, where `(h,k)` are the coordinates of the vertex.

Explanation:

The first step is to define your variables
`h=-4`
`k=2`

And we know one set of points on the graph, so
`x=-7`
`y=-34`

Next solve the formula for `a`
`y=a(x-h)^2+k`
`-34=a(-7+4)^2+2`
`-34=a(-3)^2+2`
`-34=9a+2`
`-36=9a`
`-4=a`

To create a general formula for the parabola you would put in the values for `a, h`, and `k` and then simplify.
`y=a(x-h)^2+k`
`y=-4(x+4)^2+2`
`y=-4(x^2+8x+16)+2`
`y=-4x^2-32x-64+2`

So the equation of a parabola that has a vertex at `(-4,2)` and passes through point `(-7,-34)` is:
`y=-4x^2-32x-62`