If all that is known of a plot is that the curve passes through certain squares on a grid, is there any way, other than by estimation or trial & error, to work out what the equations to the curves may be?
Please see example below:
From the filled squares, I would like to work out the maximum curve, minimum curve, & the resultant step function.
(In the above example, the max & min curves are clearly $x^2$ & $(x-1)^2$. The blue curve enters the shaded squares on the LHS & the red curve exits the shaded squares on the RHS.)
Best Answer
If the problem is to have a smooth "mean" curve, the equation is y(x)=(x-(1/2))²
If the problem is to have a stepped curve, the exact equation involves a Fourier series . An approximate is obtained with the the series limited to a number of terms depending on the expected accuracy.
$$y(x)=\left(x-\dfrac12+\dfrac1\pi\sum_{k=1}^\infty\dfrac1k\sin(2k\pi x)\right)^2$$