I'm stumped. I need the equation of a parabola that opens down, passes through the origin, passes through the point (n,0), and has vertex (n/2,h). Thanks in advance.

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- Aug 7th 2007, 10:59 AMeliminatorequation of a specific parabola
I'm stumped. I need the equation of a parabola that opens down, passes through the origin, passes through the point (n,0), and has vertex (n/2,h). Thanks in advance.

- Aug 7th 2007, 11:24 AMtopsquark
Well, the form of a vertically opening (either up or down) parabola is

where (p, q) is the vertex. (We usually use (h, k) as the vertex, but you already have an "h" in the problem.)

It opens down, so a is negative. So let's write this as

and let's make "b" positive, just to remind us.

It has a vertex so your parabola is

Now, the parabola passes through (0, 0) so...

and it passes through (n, 0), so....

So we need to solve the simultaneous equations for h and b in terms of n:

So

Thus the parabola is:

-Dan - Aug 7th 2007, 11:28 AMtopsquark
Another way to do this is to note that two of the points we have on this parabola are the x-intercepts: (0, 0) and (n, 0), so the parabola must have the form:

(The outside coefficient must be negative for it to open downward.)

We know it must pass through the vertex :

Solve this for a:

Thus the parabola is

You can verify that is indeed the vertex of this parabola.

-Dan