# Finding the equation of a curve where slope of tangent at $x=55$ is $1$

I am working on an algorithm for my work and we are using an exponential function where $x =$ the users age and $f(x) = a$ value that will be subtracted from the users compatibility index.

I havent worked with calculus in quite a while so finding an answer to this problem has been imposible. Basically I need an equation that fits the following rules.

1. vertical asymptote at $x = 100$ ($100$ is the maximum age that can be put in)
2. graph is a positive function (in the first quadrant)
3. At $x=55$ ($y$ unknown), a tangent line touches the curve; the slope of this tangent line must equal 1.
4. $f(18) = 0$

EDIT: I have another problem as well. SImilar situation.

1. vertical asymptote at $x = 18$ ($18$ is the maximum age that can be put in)
2. graph is a negative function (in the first quadrant)
3. At $x=8$ ($y$ unknown), a tangent line touches the curve; the slope of this tangent line must equal -1.
4. $f(0) = 100$
• You have problem: exponential functions do not have vertical asymptotes. Even a sum of exponential functions of the form $\sum a_ie^{b_ix}$ is finite for all real $x$. – Fly by Night Jan 23 '13 at 19:59

If you aren't worried about the function being defined when $x>100$, then you can start with the function $$g_1(x)=-\frac1x$$ to get a workable right general shape. Next, shift it to the right by $100$ to get the correct vertical asymptote--yielding $$g_2(x)=-\frac1{x-100}.$$ This satisfies the first two conditions. To satisfy the third condition, scale by a factor of $A$ to get $$g_3(x)=-\frac{A}{x-100}$$ (You'll need to set $g_3'(55)=1$ and solve to find the appropriate $A$.) Finally, shift vertically by some constant $C$ to get $$g_4(x)=-\frac{A}{x-100}+C$$ (You'll need to set $g_4(18)=0$ to solve for $C$.) Restrict $g_4$ to the interval $[0,100)$ to get the desired function $f$.
If you require that the function be defined for $x>100$, then this won't work. Let me know, if so.
Edit: Apparently, you are looking for an exponential function. That would be a function of the form $f(x)=Ae^{x-B}+C$ for some real $A,B,C$. Such functions have $y=C$ for a horizontal asymptote, but have no vertical asymptote, so such a function cannot satisfy your desired conditions.
• @Connor: Your other problem is, indeed, quite similar. We'll proceed in almost exactly the same way, but we'll drop the "$-$" sign from each of the $4$ intermediate steps (since we need a negative function), and make substitutions for each of the corresponding numbers (i.e.: $100$ should now be $18$ in $g_2$ and thereafter, $55$ should now be $8$, $1$ should be $-1$, etc.). – Cameron Buie Jan 24 '13 at 1:22