# Gaussian Function

Gaussian functions of the form
$f(x) = a e^{-\frac{(x-b)^2}{c^2}}$
and the properties of their graphs are explored.

## Properties of the Graphs of Gaussian Functions

Below are shown gauusian function with $a = 1$, $b = 0$ and different values of $c$. Note all functions of the form $f(x) = e^{-\frac{x^2}{c^2}}$ have a maximum equal to $1$ at $x = 0$. We also conclude that the parameter $c$ controls the width of the graph of $f$. As $c$ get larger, the width of the graph gets larger.

Below are shown gauusian function with $a = 1$, $c^2 = 2$ and different values of $b$. We also conclude that the parameter $b$ controls the horizontal position (or shifting) of the graph. For $b$ positive the graph is shifted to the right, and when $b$ is negative, the graph is shifted to the left.

## Gaussian Function as a Probability Density Function

The probability density function of a random variable that is normally distributed with mean $\mu$ and variance $\sigma^2$, is given by Gauusian of the form. $f(x) = \dfrac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma} \right)^2}$
and
$\int_{-\infty}^{+\infty} \dfrac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma} \right)^2} dx = 1$

## Interactive Tutorial to Further Explore Gaussian Functions

Change parameters $a$, $b$ and $c$ in the function $f(x) = a e^{-\frac{(x-b)^2}{c^2}}$ following the activities below and explain.

 a = 1 -10+10 b = 0 -10+10 c = 1 1+10
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1 - click on the button above "draw" to start. The graph of the gaussian function is displayed.

2 - Use the sliders to set parameters b to 0 and c to 1 and change parameter a. What happens to the graph?

3 - Now set parameters a to 1 and c to 1 and change parameter b. What happens to the graph? Explain analytically.

4 - Set parameters a to 1 and b to 0 and change parameter c. What happens to the graph when c takes small values? What happens to the graph when c takes larger values?

## Answers to the Above Questions

2 - The graph expands vertically as a increases. The y values of the points making the graph are multiplied by the value of a.

3 - As b increases, the graph is shifted to the right. As b decreases the graph is shifted to the left.

4 - As c is made small, the graph shrinks horizontally. As c is made large, the graph expands horizontally.