# Exponential Function Graph maker

Instructions: This Exponential Function Graph maker will allow you to plot an exponential function, or to compare two exponential functions. You need to provide the initial value $$A_0$$ and the rate $$r$$ of each of the functions of the form $$f(t) = A_0 e^{rt}$$.

Initial Value Function 1 ($$A_0$$) =
Change Rate Function 1 ($$r)$$. Ex. 0.02, 0.04, etc) =
Initial Value Function 2 (Optional. For graphing a second function) =
Change Rate Function 2 (Optional. For graphing a second function) =
Points to evaluate (Optional. Comma or space separated) =

## Exponential Function Graph Maker

This graphing tool allows you to graph one exponential function, or to compare the graph of two exponential functions. These exponential functions will have the form:

$f(t) = A_0 e^{kt}$

In order to get the graph, you just need to specify the parameters $$A_0$$ and $$k$$ for one or two functions (depending on whether you want to graph one function or if you want to compare two functions).

### 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 to graph an exponential function

An exponential function of the form that was specified above will have a characteristic exponential shape, and its general form will depend on whether the rate $$r$$ is positive or negative.

For a positive rate $$r$$ we will have exponential growth , and for a negative rate $$r$$ we will have exponential decay .

## What are the main characteristics of exponential graphs?

They have very specific shapes, as they grow or decay (depending on the sign of $$r$$) very rapidly. There are not that many types of graphs in this case. Only rapid (exponential) decay or rapid (exponential) growth.