Finding quadratic function from two points and known max value

[Nigel (UK)]

I think that there are two possible solutions, but I could be wrong too!

Let the maximum y-value be \(c\) and let this occur at \(x=b\), where \(b\) is as yet unknown. Then the parabola's equation can be written as
$$y=c-a(x-b)^2,$$
where \(a\) is another unknown.

We know two points \((x_1, y_1)\) and \((x_2, y_2)\) on the curve. Substituting these into the equation above gives
$$\eqalign{y_1=c-a(x_1-b)^2\cr
y_2=c-a(x_2-b)^2\cr}.$$
Rearranging gives
$${x_1-b\over x_2-b}=\pm\sqrt{c-y_1\over c-y_2}.$$
The value of \(b\) is between \(x_1\) and \(x_2\) if the right-hand side is negative and outside this range otherwise; this is where the ambiguity comes in. Since \(c\), \(y_1\), and \(y_2\) are known the right hand side can be computed and a sign chosen; this means that \(b\) can be found, and then either of the previous equations can be used to give \(a\).

Is this what you are looking for?

Nigel (UK)