This document discusses how to find the equation of a parabola given its focus and vertex. It explains that the distance from the vertex to the focus allows you to calculate the p term in the quadratic equation. Two examples are provided, one with the vertex at the origin and one with the vertex shifted. In the second example, the focus is (2,3) and the vertex is (2,6), so p=-3 and the equation is (x-2)2 = -12(y-2).

1. Parabolas

2. Lesson 3
Find the equation of a parabola
when we are given the coordinates
of its focus and vertex.

3. Now, we are going to begin taking what we have learned and start
piecing it together. If we are given a focus and a vertex, we have
enough to be able to generate a quadratic equation of a parabola. If
we think about it for a second, we will be able to find the distance
from the vertex to the focus based on this given information. We will
then be able to calculate our p term (the term from the previous
lesson that is in front of our non-squared variable). Placing the
coordinates of the vertex into the equation is very simple, relative to
what we have learned so far.

4. Example
Let us suppose that we are given a focus of (-6, 0) and the vertex is at the
origin.
Based on what we know without plugging anything in, we can say that the
parabola will be opening up to the left because its focus is to the left of the
origin. Now in beginning to piece things together, we can say that the
equation will be something like y2 is equal to some x term.
Since the origin is the vertex, we can say that this will be (y - 0)2
= 4p(x - 0),
which simplifies to y2 = 4px.
We know that p = -6, and we know that 4p = -24. We should now be able to
tell that the equation is y2
= -24x.

5. Example 6
We will now try a problem that has the parabola opening up or down.
We will make the focus (2, 3) and the vertex (2, 6).
The focus is directly below the vertex by 3 units; so p = -3; so 4p =
-12; but not so fast! We aren't quite home free yet. The vertex is
shifted off of the origin, and we need to consider the h and k terms.
The equation with a parabola facing downward will be (x - h)2
= 4p(y
- k), where 4p is negative. To again piece things together:
(x - 2)2
= -12 (y - 2).

6. Example 6
We will now try a problem that has the parabola opening up or down.
We will make the focus (2, 3) and the vertex (2, 6).
The focus is directly below the vertex by 3 units; so p = -3; so 4p =
-12; but not so fast! We aren't quite home free yet. The vertex is
shifted off of the origin, and we need to consider the h and k terms.
The equation with a parabola facing downward will be (x - h)2
= 4p(y
- k), where 4p is negative. To again piece things together:
(x - 2)2
= -12 (y - 2).