Definition:Logistic Curve

Definition

The logistic curve $\KK$ is the plane curve whose equation in the Cartesian plane can be given as:

$y = \dfrac k {1 + \map \exp {a - b x} }$

where $a, b, k \in \R$ are real constants.

Thus:

the lines $y = 0$ and $y = k$ are the asymptotes of $\KK$

$\KK$ has one point of inflection $p_i$, which is located at $\tuple {\dfrac a b, \dfrac k 2}$

the slope of $\KK$ at $p_i$ is $\dfrac {k b} 4$

the tangent to $\KK$ at $p_i$ is given by the equation:

$y = \dfrac k 4 \paren {b x + 2 - a}$

When $k = 0$, $\KK$ degenerates to the straight line $y = 0$.

When $b = 0$, $\KK$ degenerates to the straight line $y = \dfrac k {1 + \map \exp a}$.

The logistic curve is also known as the logistic function.

It is also known as the sigmoid curve or sigmoid function.

However, the latter term is also used for the specific instance of the logistic curve where $k = 1$, $a = 0$ and $b = 1$.