# 17.03.03 - Solving Parabola Problems (Geometry)

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It is always a good idea to work through some practice problems, so let's do this.

One obvious problem is to find the general equation from certain parameters. The simplest is if you are given the location of the vertex, the orientation of the parabola, and the parameter p.

1. Given a parabola is aligned with the x-axis, the vertex is located at (4, 9), and p = 7. Determine the following:
2. a) In which direction does this parabola open?
Solution: Since this parabola is aligned with the x-axis, and p is positive, the parabola opens to the right.

b) Find the focal point and the equation of the directrix.
Solution: Since this parabola is aligned with the x-axis, the location of the focal point is (p + h, k). Therefore, the location of this focal point is (11, 9). Also, since this parabola is aligned with the x-axis, the equation of the directrix is x = -p + h. Therefore, the equation of this directrix is

x = -3. (eq. 19)

c) What is the equation of this parabola?
Solution: The general equation for a parabola that is aligned with the x-axis is

4 p(x - h) = (y - k)2. (eq. 13)

Therefore, the equation of this parabola is

4(7)(x - 4) = (y - 9)2 (eq. 20)

28(x - 4) = (y - 9)2. (eq. 21)

d) Sketch the graph of this parabola.
Solution: We do not have a tool that will construct parabolas. Therefore, we can only approximate this curve. However, we do have a few tools to work with. We know the vertex, the focus and the directrix. Start by plotting these on the graph. Then draw the curve so that the distance between the focus and the directrix is about the same. You can find the x and y values of a few points on the curve to improve the graph.

A slightly more complicated problem is to be given the location of the focus and equation of the directrix.

3. Given the focus of a parabola is (4, -6) and the equation of the directrix is y = 3. Determine each of the following:
4. a) What is the alignment of this parabola?
Solution: Since the directrix is a horizontal line, the parabola is aligned with the y-axis.

b) Find the location of the vertex.
Solution: We know that for a parabola that is aligned with the y-axis, the focus is at (h, p + k). This focus is at (4, -6). By comparing terms we know that h = 4, and

p + k = -6. (eq. 22)

This gives us one term. We also know that for a parabola that is aligned with the y-axis, the directrix is located at y = -p + k. Our directrix is located at y = 3. Again, comparing terms we find that

-p + k = 3. (eq. 23)

We can solve the system of linear equations formed by equation 22 and equation 23 to find p and k,

p + k = -6 (eq. 22)

-p + k = 3. (eq. 23)

You can use any method you choose; I think I will add the equations,

(eq. 24a)

Solve this for k,

2 k = -3 (eq. 24a)

(eq. 25)

The vertex of any parabola is located at (h, k). Therefore, the vertex of this parabola is at (4, ).

c) Find the parameter, p of this parabola.
Solution: We have two equations for p from the previous question. We just found the value of k. Plug that into either equation, and solve for p. I will use equation 22.

p + k = -6 (eq. 22)

(eq. 26)

(eq. 27)

Therefore, the parameter p is equal to

d) In which direction does this parabola open?
Solution: We know that this parabola is aligned with the y-axis, and that p is negative. Therefore, this parabola opens downwards.

e) What is the equation of this parabola?
Solution: You know the vertex and the parameter p. You know that this parabola is aligned
with the y-axis. Therefore, you need to plug the values for the vertex and the parameter into equation 12.

4 p (y - k) = (x - h)2 (eq. 12)

(eq. 28)

(eq. 28a)

f) Sketch the graph of this parabola.
Solution: As before, we can only approximate the graph, but we know the vertex, the focus and the directrix. Plot these on the graph, and plot a couple additional points on the parabola. Then draw the curve so that the distance between the focus and the directrix is about the same.

Obviously, we should practice finding the parameters of the ellipse from the equation.

5. A certain ellipse has the equation
6. -4(x + 6) = (y + 7)2. (eq. 29)

Determine the following:

a) What is the alignment of this parabola?
Solution: Since this equation is of the form x = y2, the parabola is aligned with the x-axis.

b) What is the vertex of this parabola?
Solution: We can read the vertex from the equation. The vertex is located at (-6, -7).

c) What is the parameter, p, of this parabola?
Solution: By comparing terms we see that 4 p = -4. Solve this for p,

4 p = -4 (eq. 30)

p = -1 (eq. 31)

d) Find the location of the focal point and the equation of the directrix.
Solution: Since this parabola is aligned with the x-axis, the general location of the focal point is (p + h, k). Therefore, the location of this focal point is (-7, -7). Also, the general equation for the directrix for a parabola that is aligned with the x-axis is x = -p + h.
Therefore, the equation of this directrix is x = -5.

e) In which direction does this parabola open?
Solution: Since this parabola is aligned with the x-axis, and the parameter p is negative, this parabola opens to the left.

f) Sketch the graph of this parabola.
Solution: You know what to do. Plot the vertex, the focus and the directrix. Plot a couple additional points on the parabola. Then draw the curve so that the distance between the focus and the directrix is about the same.