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.
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:
Thus the parabola is:
(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.