# Exponential and Logistic Functions on a Graphing Calculator

## Key Questions

The carrying capacity is the limit of $P \left(t\right)$ as $t \to \infty$.

#### Explanation:

The term "carrying capacity" with respect to a logistic function is generally used when describing the population dynamics in biology. Suppose that we are trying to model the growth of a butterfly population.

We'll have some logistic function $P \left(t\right)$ which describes the number of butterflies at time $t$. In this function will be some term which describes the carrying capacity of the system, usually denoted $K = \text{carrying capacity}$.

If the number of butterflies is greater than the carrying capacity, the population will tend to shrink with time. If the number of butterflies is less than the carrying capacity, the population will tend to grow with time. If we let enough time pass, the population should tend toward the carrying capacity.

Thus, the carrying capacity can be thought of as the limit of $P \left(t\right)$ as $t \to \infty$, where $P \left(t\right)$ is a logistic growth function.

• Probably one of the most common mistakes is forgetting to put the parentheses on some functions.

For example, if I were going to graph $y = {5}^{2 x}$ as stated in a problem, some students may put in calculator 5^2x. However, the calculator reads that it is ${5}^{2} x$ and not as given. So it is important to put parentheses in and write 5^(2x).

For logistic functions, one error can involve using natural log vs. log incorrectly, like:

$y = \ln \left(2 x\right)$, which is ${e}^{y} = 2 x$; versus
$y = \log \left(2 x\right)$, which is for ${10}^{y} = 2 x$.

Exponent conversions to logistic functions may be tricky as well. If I were to graph ${2}^{y} = x$ as a y-function of x, it would be:

${\log}_{2} \left(x\right) = y$ or $\log \frac{x}{\log} \left(2\right) = y$ in calculator.

These are few of the mistakes most people tend to make. The best way to prevent this is to practice and to be careful on inputting the values so that those functions are good to graph.

If there are more mistakes that I have not mentioned, feel free to add some more.

• Like with all functions, you just need to type them out on your calculator in the $y =$ area. You will need to know that the "carrot key" (Looks like ^ , right under the "Clear" key on TI-83s and 84s) is what you use to type out your exponents. Also, especially with logistic functions, you should be sure to use parenthesis properly.

In the TI-83s and 84s, I personally find that typing out functions with lots of stuff in them (like logistic growth models or gaussian models) is rather hard to type out correctly. Therefore, if you are allowed to, I would recommend that you used another, more powerful graphing utility on a computer. Desmos.com is a good option for this.

Hope that helps :)