Question

What is the equation of the parabola that has a vertex at `(5, 2)` and passes through point `(6,9)`?

Answer 1

`f(x) = 7(x-5)^2+2`

Explanation:

Vertex form of a parabola with a vertex at `(5,2)`
`f(x) = a(x-5)^2+2`

To find the value of `a`, think about how the y increases in relation to the vertex of the parabola.

Start from the vertex, move right 1 unit. If `a = 1`, then the parabola would intersect `(5 color(blue)(+1) , 2 color(green)(+1) )`. In our case, however, the parabola must intersect `(5 color(blue)(+1) , 2 color(red)(+7) )`.

Therefore, our `a` value is equal to `frac{color(red)(7)}{color(green)(1)} = 7`

`f(x) = 7(x-5)^2+2`

graph{7(x-5)^2+2 [-2.7, 17.3, -2.21, 7.79]}