Question

How do you write the equation of the parabola in vertex form given vertex (3,3) and focus: (-2,3)?

Answer 1

The vertex form for the equation of a parabola whose focus is shifted horizontally a signed distance, `f`, from its vertex is:

`x=1/(4f)(y-k)^2+h" [1]"`

where `(h,k)` is the vertex.

Explanation:

Substitute the vertex `(3,3)` into equation [1]:

`x=1/(4f)(y-3)^2+3" [2]"`

Compute the signed distance from the vertex to the focus:

`f = -2 - 3`

`f = -5`

Substitute `f=-5` into equation [2]:

`x=1/(4(-5))(y-3)^2+3`

Simplify the denominator:

`x=-1/20(y-3)^2+3`

Answer 2

`x = -1/20(y-3)^2+3`

Explanation:

First, let's figure out which way we need to draw the parabola.

The vertex is (3,3) and the focus is (-2,3).

These points have the same y-value, so they form a horizontal line. This means that the parabola will be horizontal.

It must be of the form `x = a(y-k)^2+h`

Additionally, the focus is to the LEFT of the vertex, so the parabola will point to the LEFT (meaning `a` is negative).

Let's call the distance between the focus and the vertex `c`.

We know that the value of `a` is equal to `+-1/(4c)`.

In this case, `c` is 5, since the vertex and focus are `5` units apart.

`a = +-1/(4(5)) = +- 1/20 = -1/20` since we already know `a` is negative.

This gives us everything we need to write our parabola's equation! The vertex `(h,k)` is (3,3), the stretch factor `a` is -1/20, and the direction is horizontal.

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

`x = -1/20(y-3)^2+3`

Final Answer