I don't understand how to write the equations.

Question

A parabola can be drawn given a focus of (7,6) and a directrix of y=8. Write the equation of the parabola in any form.

Answer 1

Let's use (x - h)² = p(y -k).

The vertex is halfway between your focus and directrix, so...

(8 + 6)/2 = 7

The vertex is (7,7), and p is 8 - 7 = 1. Since we don't write out 1 times anything...

(x - 7)² = y - 7

Answer 2

All about parabolas:

https://www.mathsisfun.com/geometry/parabola.html

If the origin is at (0,0), then a vertical parabola is x² = 4ay. Since you have a directrix at y = 8, you have a vertical form.

In the general formula, "a" is the distance from the origin to either the directrix or the focus. For your case, a=1 (the directrix is 2 units away from the focus, and the origin is halfway between) Thus, the origin is at (7,7)

Since the directrix is above the origin, a is negative

The origin at (7,7) means that x gets replaced by x-7, and y gets replaced by y-7.

SO:

-4(y - 7) = (x - 7)²

To help visualize, plot this using an online graphing tool---or a graphing calculator

Answer 3

This problem involves a merging of two approaches to a parabola: parabola as graph of a quadratic equation and parabola as a conic section.

The quadratic equation approach has two basic forms:

Standard form y = ax² + bx + c, and

Vertex form y = a(x - h)² + k, where (h, k) is the vertex.

When treated as a conic section, the focus and directrix must be taken into account. The only effect this has is to replace a with the expression 1/(4c) to account for the focus/directrix.

But first we need to find the vertex from the clues in the question. We are told that the directrix is y = 8. This is a horizontal line 8 units above the x-axis. This means that the parabola's line if symmetry is vertical and is therefore a "y =" parabola, rather than an "x =" parabola. The vertex is on the axis of symmetry exactly halfway between the focus and the directrix. Since the focus is at (7, 6), then the vertex will be at (7,7) because 7 is between the y-values 6 and 8. From this, we can put together most of vertex form: y = a(x - 7)² + 7.

We also find out that this is a downward-facing parabola, since the focus is below the directrix. This means that the a or 1/(4c) multiplier will be negative. The c value is the distance between the vertex and the focus, which is 1. So this gives us the last piece of the puzzle, that the multiplier is -1/4.

y = -(1/4)(x - 7)² + 7.

Answer 4

All about parabolas:

https://www.mathsisfun.com/geometry/parabola.html

If the origin is at (0,0), then a vertical parabola is x² = 4ay. Since you have a directrix at y = 8, you have a vertical form.

In the general formula, "a" is the distance from the origin to either the directrix or the focus. For your case, a=1 (the directrix is 2 units away from the focus, and the origin is halfway between) Thus, the origin is at (7,7)

Since the directrix is above the origin, a is negative

The origin at (7,7) means that x gets replaced by x-7, and y gets replaced by y-7.

SO:

-4(y - 7) = (x - 7)²

To help visualize, plot this using an online graphing tool---or a graphing calculator