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:


(y-y_0) = \frac 1 {4p} (x-x_0)^2


where the vertex is at (x_0,y_0) 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 x_0 is 0. So:


y = \frac 1 {4p} x^2 + y_0


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


-2 =( \frac 1 {4p} ) 4^2 + y_0


You need one more data point to determine the value for p and y_0. 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:


y = \frac {x^2} 4 - 6
with vertex at (0,-6) and focus at (0,-5), or


y = \frac {x^2} 2 - 10
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.