Gaussian
functions of the form
\[ f(x) = a e^{-\frac{(x-b)^2}{c^2}} \]
and the properties of their graphs are explored.
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.
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
\( \int_{-\infty}^{+\infty} \dfrac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma} \right)^2} dx = 1 \)
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.