Question

What is the equation of the parabola that has a vertex at `(2, -1)` and passes through point `(3,-4)`?

Answer 1

`y=-3x^2+12x-13`

Explanation:

This problem will be easier if we express the equation for the parabola in vertex form.

`y=a(x-k)^2+h`

where `k` is the `x`-coordinate of the vertex and `h` is the `y`-coordinate of the vertex. Since the vertex is at `(2, -1)` our equation for the parabola becomes

`y=a(x-2)^2-1`

Since the parabola passes through the point `(3, -4)` we can write

`-4=a(3-2)^2-1=a-1`

and solve for `a` by adding 1 to both sides.

`a=-3`

So the equation for the parabola is

`y=-3(x-2)^2-1`.

We can expand the binomial to obtain the equation for the parabola in standard form.

`y=-3(x^2-4x+4)-1=-3x^2+12x-13`

graph{-3x^2+12x-13 [-1, 4, -10, 5]}