Parabolas

Lesson 3

Find the equation of a parabolawhen we are given the

coordinatesof its focus and vertex.

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.

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.

Example 6We 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).