Aradio telescope has a parabolic surface as shown below.

a parabola opening up with vertex at the origin is graphed on the coordinate plane. the height of the parabola from top to bottom is 9 meters and its width from left to right is 12 meters.

if the telescope is 9 m deep and 12 m wide, how far is the focus from the vertex?

OK, so the graph is a parabola, with points x=0,y=0; x=6,y=-9; and x=12,y=0 

Because the roots of the equation are 0 and 12, we know the formula is therefore of the form 
y = ax(x - 12), for some a 

So put in x = 6 
-9 = 6a(-6) 
9 = 36a 
a = 1/4 

So the parabola has a curve y = x(x-12) / 4, which can also be written y = 0.25x² - 3x 

The gradient of this is dy/dx = 0.5x - 3 

The key property of a parabolic dish is that it focuses radio waves travelling parallel to the y axis to a single point. So we should arrive at the same focal point no matter what point we chose to look at. So we can pick any point we like - e.g. the point x = 4, y = -8 

Gradient of the parabolic mirror at x = 4 is -1 
So the gradient of the normal to the mirror at x = 4 is therefore 1. 

Radio waves initially travelling vertically downwards are reflected about the normal - which has a gradient of 1, so they're reflected so that they are travelling horizontally. So they arrive parallel to the y axis, and leave parallel to the x axis. 

So the focal point is at y = -8, i.e. 1 metre above the back of the dish.

Hello there.

A radio telescope has a parabolic surface as shown below.

A parabola opening up with vertex at the origin is graphed on the coordinate plane. The height of the parabola from top to bottom is 9 meters and its width from left to right is 12 meters. 

If the telescope is 9 m deep and 12 m wide, how far is the focus from the vertex?

 y = -8, i.e. 1