How can I eliminate constants k and c without using a number?

Question

y = k(x-a)^2 + c

I need to use this equation to find the moment of inertia and express my answer in terms of a number and powers of a and b? Below is the image of the figure used.

http://postimage.org/image/ojjgac35x/

Someone please help!!! If I can figure out how to eliminate the constants k and c, I could figure out the rest. I'm not given anything else. No density, lengths, or heights.

Answer 1

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Notice that y = k(x - a)^2 + c has its vertex at x = a, so the point (a, b) is the parabola's vertex (note that c = b since (a, b) lies on the parabola and, thus, must be its vertex). So, at this point, we have:

y = k(x - a)^2 + b.

Then, from the graph, (0, 0) lies on the parabola, so plugging in x = y = 0 yields:

0 = k(0 - a)^2 + b = ka^2 + b ==> k = -b/a^2.

Thus, the equation of the parabola in terms of a and b only is:

y = (-b/a^2)(x - a)^2 + b.

I hope this helps!