1. derive the equation of the parabola with a focus at (4, −7) and a directrix of y = −15. put the equation in standard form. 2. derive the equation of the parabola with a focus at (0, 1) and a directrix of y = −1. 3. derive the equation of the parabola with a focus at (−7, 5) and a directrix of y = −11. 4. derive the equation of the parabola with a focus at (2, −1) and a directrix of y = −1/2.

To derive the equation of the parabola, let (x , y) be a point in the parabola. Its distance from the focus should be equal to its distance from the directrix.

I'll show here how to do number 1.
1.)  focus : (4, -7)    and directrix of y = -15
                       d (point to focus) = d (point to directrix)
                         sqrt ((x - 4)² + (y + 7)²) = (y + 15)
Squaring both sides gives us,
                                  (x - 4)² + (y + 7)² = (y + 15)²
Simplifying gives,
                       x² - 8x + 16 + y² + 14y + 49 = y² + 30y + 225
Simplifying leads to,
                                        16y = x² - 8x -160
This leads to our final answer of
                                   y = (1/16)x² - (1/2)x - 10

All the rest of the numbers follow the same steps to get answered.