Hi, Kirsten.

The vertex form equation for a parabola is y = (x - h)^{2} + k, where (h,k) is the vertex.

In this case, we have to complete the square on the equation to put it in this form. This gets a little annoying when x has a coefficient, but it's even more annoying in this case because the coefficient isn't factored easily from the rest of the terms.

First, move the constant out of the way (some people leave it on the same side; I tend to move it over so I don't have to think about it for a bit)...

f(x) - 4= -9x^{2} + 5x

To complete the square, the coefficient of x must be 1. So we have to factor a -9 out on the right side.

f(x) - 4 = -9[x^{2} - (5/9)x]

Completing the square involves taking half the middle term's coefficient and squaring it, then adding the result and adjusting the equation accordingly. In this case, half the middle term is 5/18; squaring that gives us 25/324. Not a happy number, but we'll deal with it.

So to complete the square, we have to add 25/324 INSIDE the parentheses on the right side. But because it's inside the parentheses, we aren't really adding 25/324... we're adding -9 • 25/324. So we are actually adding -225/324, or -25/36. Since we are adding this on one side of the equation, we must also add it on the other to keep things balanced. So...

f(x) - 4 -25/36 = -9[x^{2} - (5/9)x + 25/324]

Factoring the right side and combining constants on the left....

f(x) - 169/36 = -9[x^{2} - (5/18)]^{2}

Moving the constant back over, we have...

f(x) = -9[x^{2} - (5/18)]^{2} + 169/36.

Our vertex therefore is (5/18, 169/36).

Now, since you only had to find the x-coordinate, you could really stop when you think about half the middle term with the -9 factored out. In other words, once I see that the middle term is -5/9, I can tell that the x-coordinate is 5/18 because I know the added constant inside the factored parentheses will be half that middle term. But it's good to see the whole thing for a full understanding.