Question

What is the standard form of the parabola with a vertex at (4,0) and a focus at (4,-4)?

Answer 1

`y = -1/16(x - 4)^2`

Explanation:

The standard form of a parabola is

`y = 1/(4p)(x - h)^2 + k`

where `(h, k)` is the vertex and `p` is the distance from the vertex to the focus (or the distance from the vertex to the directrix).

Since we are given the vertex `(4, 0)`, we can plug this into our parabola formula.

`y = 1/(4p)(x - 4)^2 + 0`

`y = 1/(4p)(x - 4)^2`

To help visualize `p`, let's plot our given points on a graph.

`p`, or the distance from the vertex to the focus, is -4. Plug this value into the equation:

`y = 1/(4(-4))(x - 4)^2`
`y = -1/16(x - 4)^2`

That's your parabola in standard form!