Algebra 2 focus and directrix

Question

A parobala has a focus of (0,3) and a directrix of y=5. Write an equation for the parobala in vertex form.

Answer 1

Hello Sarit,

First, in case you haven't done it yet, it's helpful to sketch the info you're given. Some background info (or reminders):

1. the vertex always sits HALFWAY between the directrix and the focus
2. a parabola always opens toward its focus
3. if the directrix is vertical (x = some constant), the parabola is horizontal (opens left OR right)
4. if the directrix is horizontal (y = some constant), the parabola is vertical ( opens up OR down)

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The focus is 2 units away from the directrix. So, the vertex will be one unit away from either of them, with coordinates (0,4) where h = 0 and k = 4

You will notice y = 5 means the directrix is horizontal. So we can use the equation 4p (y - k) = (x-h)²

where p = distance FROM vertex to focus.

From vertex to focus, you go 1 unit DOWN, so p = -1

So your equation is 4(-1)(y - 4) = (x-0)² which is -4(y - 4) = x² which we can write as y-4 = (-1/4)x²

Add 4 to both sides: y = (-1/4)x² + 4



Answer: y = - (1/4)x² + 4