Finding quadratic function from two points and known max value
08-31-2017, 02:46 PM
Post: #4
 Nigel (UK) Senior Member Posts: 472 Joined: Dec 2013
RE: Finding quadratic function from two points and known max value
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)
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 Messages In This Thread Finding quadratic function from two points and known max value - Dave Britten - 08-31-2017, 12:57 PM RE: Finding quadratic function from two points and known max value - grsbanks - 08-31-2017, 01:03 PM RE: Finding quadratic function from two points and known max value - Dave Britten - 08-31-2017, 01:09 PM RE: Finding quadratic function from two points and known max value - Nigel (UK) - 08-31-2017 02:46 PM RE: Finding quadratic function from two points and known max value - Dave Britten - 08-31-2017, 05:02 PM RE: Finding quadratic function from two points and known max value - Dave Britten - 08-31-2017, 05:31 PM

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