Force Graph A into Graph B with Same Area But Diff Shape

[jdlewis5293]

So here is the problem I am trying to figure out. This is for an Sociology Independent study, so this isn't a homework assignment, but for the greater good of humanity. ;)

First a few "rules".

-There is Graph A.
-The x-axis is 100 units long, 1 being the left end, 100 being the right.
-It is very shallow on the left end (at point 1 on the x-axis it is only 1 unit tall (1 unit up on the y axis).
-It is very tall on the right end (at point 100 on the x-axis, it is about 150 units tall (150 units up on the y-axis).
-The line graph is a curve, the slope is not consistent throughout. The slope isn't very steep in the beginning/middle, but is very steep at the end.

-There is Graph B.
-The x-axis is 100 units long, 1 being the left end, 100 being the right. (same as Graph A)
-It is shallow, but not as much as graph A (at point 1 on the x-axis, it is about 4 units tall)
-It is tall on the right end(at point 100 on the x-axis, it is about 80 units tall)
-The line is still a curve, the slope is still not consistent throughout, it is just less steep than graph A.

The PROBLEM:
If I were to take Graph A, and force it into the same shape as Graph B keeping the same area (graph A and B have the same area), what would the most efficient way to find the value each Y-value would have to be multiplied by in Graph A to get to Graph B.

Example 1: x-axis point 1 in Graph A would have a Y value of 1, in graph B it would be 4, therefore the value needed to turn Graph A's value to Graph B's value would be 4.

Example 2: x-axis point 100 in Graph A would have a Y value of 150, in graph B it would be 80, therefore the value needed to turn Graph A's value to Graph B's value would be .53.

By hand, I would essentially have to do every single one of these points by hand, which is unrealistic, and what if I wanted to do 1000 points instead of 100? There must be some quicker way of doing this.

Also, I use excel frequently, so if there is a quicker way by using something in excel I am interested as well. Pictures posted below!

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[mmwave]

Thank you for sharing your problem and providing clear descriptions and examples. I can understand your concern about finding an efficient way to determine the values needed to transform Graph A into the same shape as Graph B while maintaining the same area.

One solution to this problem could be to use a mathematical technique called interpolation. Interpolation is a method used to estimate values between known data points. In this case, we can use interpolation to estimate the values that would need to be multiplied to each Y-value in Graph A to transform it into Graph B.

There are different types of interpolation methods, but one that could work well for this problem is called linear interpolation. This method involves creating a straight line between two known data points and using that line to estimate the values between them.

To apply this method to your problem, you could first plot the points from Graph A and B on a graphing calculator or software such as Excel. Then, using the linear interpolation function, you can estimate the values needed for each Y-value in Graph A to match those in Graph B.

Another option could be to use a curve fitting tool in Excel. This tool allows you to fit a curve to your data points and then use that curve to estimate the Y-values for any given X-value. This could be a more accurate approach compared to linear interpolation, as it takes into account the curve in your data.

I hope these suggestions can help you find a quicker and more efficient way to solve your problem. Good luck with your independent study and for the greater good of humanity.