SOLUTION: I have to determine the foci and vertices of a conic: y squared=4(x+2y)

Question 705954: I have to determine the foci and vertices of a conic:
y squared=4(x+2y)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
With one squared term (and no xy terms) this is the equation of a parabola. Parabolas have only one focus and one vertex.

To find the vertex and focus we'll start by transforming the equation into vertex form. Different books and teachers teach slightly different versions of the vertex form of a parabola. The one I prefer is:
for vertical parabolas
or
for horizontal parabolas
With this variation of the vertex form, the "p" represents the distance between the vertex and the focus.

Since we have a term, we have a horizontal parabola. So we will be using:

And we will start by transforming

into the vertex form. First we simplify:

Next we will gather the y terms together so we can complete the square. Subtracting 8y we get:

To complete the square we take 1/2 of the coefficient of y, 1/2 of 8 is 4, and add the square of this, 4 squared is 16, to each side:

With this step the left side now matches the pattern for a perfect square, , with the "a" being "y" and the "b" being "4". So we can rewrite the left side as a perfect square:

The left side now matches the pattern of the left side of the vertex form. Now we just have to match the right side. The right side of the vertex form, 4p(x-h), is a factored expression with just plain x in the parentheses. To get this form we will factor out x's coefficient from 4x+16:

The form has a subtraction in the parentheses. So we need to rewrite the addition we now have as a subtraction:

And finally the form has "4p" in front of the parentheses. We current have just "4". What number must our "p" be in order for 4p to be a 4? The answer is: 1. So now we finally have the vertex form:

We are now ready to find the vertex and focus. The "h" and "k" of the vertex form are the x and y coordinates, respectively, of the vertex. Our equation has a -4 in the position of the "h" and a "4" in the place of the "k". So:
Vertex: (-4, 4)
The "p" of the form tells us how far it is from the vertex to the focus. Our "p" is 1 so the focus is 1 unit away from the vertex. But in which direction? Since the equation is the equation of a horizontal parabola the focus will be to the right or left of the vertex. And since the "p" is positive 1 we will go in the positive horizontal direction, to the right. So we add the "p" to the "h":
Focus: (-4+1, 4) or simply (-3, 4)
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