Question

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

Answer 1

When given, `y(x) = ax^2 + bx + c`
`f = 1/(4a)`
`h = -b/(2a)`
`k = y(h)`
The focus is the point `(h, k + f)`
The equation of the directrix is `y = k - f`

Explanation:

Given: `y(x) = x^2 -2x`

`a = 1, b = -2, and c = 0`

`f = 1/(4(1))`

`f = 1/4`

`h = - (-2)/(2(1))`

`h = 1`

`k = y(1)`

`k = 1^2 - 2(1)`

`k = -1`

The focus is the point, `(1, -1 + 1/4) = (1, -3/4)`

The equation of the directrix is:

`y = -1 - 1/4`

`y = -5/4`

Here is a graph of the parabola, the focus and the directrix.