Finding quadratic function from two points and known max value

08312017, 02:46 PM
Post: #4




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 yvalue be \(c\) and let this occur at \(x=b\), where \(b\) is as yet unknown. Then the parabola's equation can be written as $$y=ca(xb)^2,$$ 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 $$\eqalign{y_1=ca(x_1b)^2\cr y_2=ca(x_2b)^2\cr}.$$ Rearranging gives $${x_1b\over x_2b}=\pm\sqrt{cy_1\over cy_2}.$$ The value of \(b\) is between \(x_1\) and \(x_2\) if the righthand 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  08312017, 12:57 PM
RE: Finding quadratic function from two points and known max value  grsbanks  08312017, 01:03 PM
RE: Finding quadratic function from two points and known max value  Dave Britten  08312017, 01:09 PM
RE: Finding quadratic function from two points and known max value  Nigel (UK)  08312017 02:46 PM
RE: Finding quadratic function from two points and known max value  Dave Britten  08312017, 05:02 PM
RE: Finding quadratic function from two points and known max value  Dave Britten  08312017, 05:31 PM

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