# Exponential Function Calculator

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. Type $$t_1$$ (One numeric expression) = Type $$y_1$$ (One numeric expression) = Type $$t_2$$ (One numeric expression) = Type $$y_2$$ (One numeric expression) = Points to evaluate (Optional. Comma or space separated) =

## 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.