The first thing we need to understand about parabolas is that they are mathematically defined as all points that are equidistant (the same distance) from a point we will call the FOCUS and a line we will call the DIRECTRIX. The point is always located in the curve of the parabola and the directrix is always on the other side of the vertex draw either vertical or horizontal so it will never intersect the parabola. See the four graphs below.
Also, for any parabola, the distance between the vertex and the focus, and the distance between the vertex and the directrix is a special distance which we will refer to as "c". Using the value for c will give us a in the following formula: a = 1/(4c). Again remember to make c negative if the parabola opens in the negative direction which also makes a negative.
Next we must understand that there are two standard formulas for parabolas. They are:
y - k = a(x - h)2
This form is used when the parabola opens up or down. (h, k) is the vertex, "a" determines whether it opens up or down. If a is positive the parabola opens up and if it is negative, the parabola opens down.
x - h = a(y - k) 2
This form is used when the parabola opens right or left. (h, k) is still the vertex....make sure you keep h with the x variable and k with the y variable. If "a" is positive the parabola opens to the right (in the positive direction) and if "a" is negative it opens left (in the negative direction.)
Remember also, in either case, the line that cuts the parabola in half is called the axis of symmetry because the parabola is symmetrical with respect to that line. If the parabola opens up or down, the axis will be a vertical line making it in the form x = h where h is the x coordinate of the vertex. If the parabola opens sideways, the axis will be a horizontal line making it in the form y = k where k is the y coordinate of the vertex.
Ok, I think we are ready to try some examples.
Lets take the case where we are given the focus F(4, -3) and directrix is y = 1. We are going to use this information to write the equation of the parabola that is formed. The first thing we should do is graph the information and make a quick sketch of the parabola so we can find the 3 pieces of missing information...the vertex and c so we can find a. We also need to make sure we are plugging these numbers into the appropriate formula depending on which way the parabola opens. So see if you can figure that out by graphing the information and then check down below.
This is what your sketch should look like:
From this sketch you should see that the vertex is halfway between the focus and the directrix making it the point V(4, -1). We should be able to determine that c = -2 making the parabola open down. So now we should be able to determine a by using the formula
a = (1/4c) and then pick the appropriate formula for this particular parabola and plug in our vertex and the value of a. Try it and check your answer below.
Since a = (1/4(-2)), a= (-1/8). Substituting this value along with the vertex gives us this equation:
y - k = a(x - h)2
y - (-1) = (-1/8)(x - 4)2
y + 1 = (-1/8)(x -4) 2
Now lets try one more with a little different given information. Suppose this time we are given the vertex V(-3,1) and the focus F(-2, 1). Lets sketch this and find the vertex and a again so we can write our equation. See what you can do and then check down below.
You should have graphed the two points to determine which way the parabola opens. Also, this is an easy way to determine c since it is the distance between the focus and the vertex. In this case the parabola opens to the right and c = 1. Therefore a = (1/4) and we use the formula x - h = a(y - k)2 since it opens sideways. Substituting the vertex for h and k and 1/4 for a we should get the final equation of:
x + 3 = 1/4(y - 1)2