I can't seem to figure out how to find the focus using the vertex or the parabola.

Problem:

Axis is the y-axis, and the parabola passes through the point (4, -2)

Thanks!

You need to provide more information to solve this. The general form for a parabola is:



where the vertex is at and the distance from vertex to focus is p. You said that "axis is the y-axis," which I assume menas that the vertex is on the y axis, but you don't know where on the y-axis . Hence you know that is 0. So:



If the parabola passes through (4,-2), you can substitute these values into the formula:



You need one more data point to determine the value for p and . Without that, p could be anything. For example if p = 1 then y_0 = -6. If p = 1/2 then y_0 = -10. So both of these equations fit the data you've provided:

with vertex at (0,-6) and focus at (0,-5), or

with vertex at (0, -10) and focus at (0, -9.5)

There are infintely many combinations of p and y_0 that work for the data you've provided.