Logistics functions of the form
\[ f(x) = \dfrac{a}{1 + b \; e^{\left( - \frac{x - c}{d} \right) } } \]are explored along with the properties of their graphs. Parameters \( a \), \( b \), \( c \), and \( d \) defining function \( f \) are changed and their effects analyzed.
Tutorial and Interactive Explorer
Tutorial Questions and Answers
- Click on the button above "draw" to display the graph of the logistics function.
- What happens to the graph of the function as \( x \) increases? Change parameter \( a \) and see what happens. Find an analytical explanation.
- Set parameters \( a \), \( b \), and \( d \) to constant values and change \( c \). What happens to the graph? Find an analytical explanation.
- Set parameters to constant values and change \( d \). What happens to the graphs? Find an analytical explanation.
Answer
As \( x \) starts from negative values and increases, \( f(x) \) starts from values close to zero then approaches a constant value equal to parameter \( a \). \( y = a \) is a horizontal asymptote.
Answer
The whole graph is shifted to the right when \( c \) increases and to the left when \( c \) decreases.
Answer
For small values of \( d \), the graph changes rapidly. For larger values of \( d \), the graph changes slowly.