Narrowing the gap between two curves

Question

I want to raise the lower graph toward the upper graph to make the points of the lower graph as close as possible to those of the upper graph, in other words, to make the graphs "match" as closely as possible.

Any idea of how to do this in mathematical way? I tried taking the means of both graph then translating the lower graph by the percentage difference of the means, but this doesn't give me what I want.

Answer 1

I believe you want to vertically translate the blue curve such that the signed area between the two curves is zero. This is equivalent to translating the blue curve upwards by the average value of the difference between the two curves. To do this:

• if the red and blue curves respectively represent $y=f(x)$ and $y=g(x)$ and the horizontal interval of interest is $[a,b],$ then translate the blue curve upwards by $$\frac1{b-a}\int_a^b f(x)-g(x)\,\mathrm dx.\tag1$$

For example,

on the interval $[2,4],$ translating this blue curve upwards by $0.5$ units (obtained from evaluation $(1)$) gives the green curve, which is its closest "match" to this red curve.

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Addendum

Thank you so much, @ryang! That worked perfectly. How would I change it from "units shifted" to percentage shifted?

• The resulting average percentage increase of $g(x)$ is $$\frac1{(b-a)^2}\int_a^bf(x)-g(x)\,\mathrm dx\int_a^b\frac1{|g(x)|}\,\mathrm dx.$$

In the above example, this equals $59.3\%.$