Question

How do you find the vertex, focus and sketch `y^2=3x`?

Answer 1

Vertex: `(0, 0)`
Directrix: `x=-3//4`
Focus: `(3/4, 0)`

Explanation:

When the `y` term is squared, the parabola opens to the left (or right). When a parabola can be expressed as

`y^2=4px`

The vertex is at `(0, 0)`

The directrix is at `x=-p`

The focus is at `(p, 0)`

The given equation, `y^2=3x`, is almost there, but the number in front of `x` doesn't have a `4` in there. What do we do?

We want the `3=4p`. Notice if I divide both sides of this equation aby `4`, you get

`p=3/4`

This allows us to rewrite the original as

`y^2=4(3/4)x`

Therefore,
Vertex: `(0, 0)`
Directrix: `x=-3//4`
Focus: `(3/4, 0)`

graph{(y^2-3x)(x+3/4)=0 [-1.2, 2, -3, 3]}