Hello cubew00t

Welcome to Math Help Forum! Originally Posted by

**cubew00t** Hello, nice to meet you all, I've been working on a spreadsheet that requires me to create a function to determine a value from a exponential curve where the area of the curve is variable . The terminal points, 0,0 and 100,100 remain constant and the curve is symmetrical at the 45degree angle. How can I derive a function knowing only these things. Thanks!

can this be expressed mathematically to solve for x and y ?

I have a solution for you that seems to satisfy all your requirements including symmetry about the line $\displaystyle x+y =100$. It is:$\displaystyle y = \frac{-kx}{x-(100+k)},\; k > 0,\; 0\le x\le 100$

The area under the curve is:$\displaystyle \int^{100}_0\frac{-kx}{x-(100+k)}dx = k(100+k)\ln\left(\frac{100+k}{k}\right)-100k$

Here are a few sample values.$\displaystyle k=1$; area = $\displaystyle 3.7$%

$\displaystyle k=10$; area = $\displaystyle 16.4$%

$\displaystyle k= 50$; area = $\displaystyle 32.4$%

The curve is not, of course, exponential. It's part of a rectangular hyperbola, with asymptotes $\displaystyle x = 100+k$ and $\displaystyle y = -k$. The curve contains the points $\displaystyle (0,0)$ and $\displaystyle (100,100)$ and is symmetrical about the line $\displaystyle x+y=100$. (You can check this by replacing $\displaystyle x$ by $\displaystyle 100-x$ in the function: the same values of $\displaystyle y$ will be produced in the reverse order.)

I attach a screen shot of the graphs for $\displaystyle k = 1, 10, 50$ and an Excel spreadsheet with the calculations that produced these graphs.

Grandad