When we are given the equation of a parabola, we must first get it into one of the two forms of a parabola

Either:

y - k = a(x - h)² (remember this one opens up or down if the x is squared)

or

x - h = a(y - k)² (remember this one opens sideways if the y is squared.)

Also, remember a = 1/4c so we have to set a = to the number in front of the paranthese to find c so we can graph the vertex and then determine where c has to go by which variable is squared, thus being able to determine the coordinates of the focus and the equation of the directrix. Lets explore what we mean.

Lets take the equation: y² + 2x = 0. To get it into the correct standard form, we must have the variable squared on one side of the equals sign and the other variable on the opposite side. So the first thing we should do is subtract the 2x to the other side. This gives us: y² = -2x....then I would switch sides for both so the square term is on the right side like in the two formulas. Remember we want to make it look like one of the two forms above. So now we should have -2x = y² . The next thing we should see from the formulas is that we don't want a number in front of the variable that is not squared, but we want one in front of the one that is squared, so divide by -2, but write it as a fraction in front of the y squared term. So we finally get the form: x = (-1/2)y².

Can you tell what the vertex is now and can you solve for c? Try it and check below.

The vertex should be (0, 0) since there is no number being subtracted from the x or y so you can think of it like this: x - 0 = (-1/2)(y - 0)² . The number in front of the parantheses is -1/2 so if you set it equal to 1/4c and solve for c you get:

  -1           1
 -----    =  -----
   2           4c

making -4c = 2 and c = -1/2

Now take this information and put it on a graph and sketch which way the parabola opens to decide where the focus will go and where the directrix will go so we can determine the equations of each. Try it and check below.

The parabola should open sideways since the y is squared so the focus must be inside the parabola a half a unit to the left since it opens to the left because a and c are negative. Look on the graph below:

This makes the focus the point (-1/2, 0) and the directrix is the vertical line x = 1/2. Now we should also be able to see that the line that creates two equal parts of the parabola is the horizontal line going through the vertex, so the axis of symmetry is the line: y =0. And now you have all the information that we were asked to find.

Lets try it on a problem that we have to complete the square to get the parabola in the correct form to find all the 4 pieces of information. We recognize that we have to complete the square because there is a variable squared and that same variable that is not squared.

Take the equation:

     x² - 6x - 4y +5 = 0

We must first get the x's on one side and the y's and constants on the other side of the equal sign. Then switch sides again so we have the squared term on the right side.....

     x² - 6x + ____ = 4y - 5 + ____
     4y - 5 +_____ = x² - 6x + ____

Then we complete the square again and divide by the coefficient in front of the y because we don't want a number in front of the variable not squared, but we do in front of the paranthese of the variable that is squared.

Divide the middle term by 2 and square it and add to both sides:

      4y - 5 + 9 = x² - 6x + 9

Simplify the left side and factor the right side:

      4y + 4 = (x - 3)²

Divide by the 4 to get the y with a coefficient of 1:

      y+ 1 = (1/4)(x - 3)²

Now we are ready to find the vertex and c so we can graph it, determine which way the parabola opens and place the focus and directrix in the correct places so that we can write the coordinates of the focus and the equations of both the directrix line and the axis. Try it and then check down below.

The vertex is (3, -1) and since a = 1/4 and a = (1/4c) c = 1. Since the x is the variable squared and the a is positive, this parabola opens up, placing the focus inside the parabola above the vertex and the directrix a horizontal line below the vertex as shown below.

Therefore the coordinates of the focus are (3, 0), the directrix is the equation: y = -2 and the axis of symmetry is the vertical line: x = 3 since that is the line that cuts the parabola into two equal pieces.