Find the equation of the exponential function

Instructions: Use this step-by-step Exponential Function Calculator, to find the function that describe the exponential function that passes through two given points in the plane XY. You need to provide the points \((t_1, y_1)\) and \((t_2, y_2)\), and this calculator will estimate the appropriate exponential function and will provide its graph.

Exponential Function Calculator from Two Points

The idea of this calculator is to estimate the parameters \(A_0\) and \(k\) for the function \(f(t)\) defined as:

\[f(t) = A_0 e^{kt}\]

so that this function passes through the given points \((t_1, y_1)\) and \((t_2, y_2)\).

But, how do you find an exponential function from points?

Technically, in order to find the parameters you need to solve the following system of equations:

\[y_1 = A_0 e^{k t_1}\] \[y_2 = A_0 e^{k t_2}\]

Solving this system for \(A_0\) and \(k\) will lead to a unique solution, provided that \(t_1 = \not t_2\).

Indeed, by dividing both sides of the equations:

\[\displaystyle \frac{y_1}{y_2} = \frac{e^{k t_1}}{e^{k t_2}}\] \[\displaystyle \Rightarrow \, \frac{y_1}{y_2} = e^{k (t_1-t_2)}\] \[\displaystyle \Rightarrow \, \ln\left(\frac{y_1}{y_2}\right) = k (t_1-t_2)\] \[\displaystyle \Rightarrow \, k = \frac{1}{t_1-t_2} \ln\left(\frac{y_1}{y_2}\right)\]

In order to solve for \(A_0\) we notice from the first equation that:

\[A_0 = y_1 e^{-k t_1} = y_1 \frac{y_2}{y_1 e^{k t_2}} =\frac{y_2}{e^{k t_2}} \]

How do you calculate exponential growth?

It is not always growth. Indeed, if the parameter \(k\) is positive, then we have exponential growth, but if the parameter \(k\) is negative, then we have exponential decay.

The parameter \(k\) will be zero only if \(y_1 = y_2\) (the two points have the same height).

For specific exponential behaviors you can check our exponential growth calculator and the exponential decay calculator , which use specific parameters for that kinds of exponential behavior.