Graph of the logistic function (sigmoid)

Q: What is the range of the logistic function?

0 < y < 1. The values 0 and 1 are asymptotes, approached but never reached.

Q: Where is the logistic curve steepest?

At its midpoint (0, ½), which is also its inflection point. The slope there is ¼.

Q: What is the logistic growth model?

P(t) = L / (1 + e^(−k(t − t₀))): growth that starts exponentially and levels off at the carrying capacity L.

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The logistic function, or sigmoid, rises in an S-shape from 0 to 1. It grows almost exponentially at first, slows down, and levels off at 1, which makes it the standard model for growth that runs into a limit.

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The graph of

y=11+e^−x

for −8 ≤ x ≤ 8. Dots mark its y-intercept.

Key features

Domain: All real numbers
Range: 0 < y < 1
Horizontal asymptotes: y = 0 as x → −∞ and y = 1 as x → ∞
Midpoint and inflection point: (0, 1/2), where the slope is greatest (1/4)
Symmetry: σ(−x) = 1 − σ(x): point symmetry about (0, ½)
Derivative: σ′(x) = σ(x)(1 − σ(x))
y-intercept: (0, 1/2)
Derivative: e^−x(1+e^−x)²

Derivative and antiderivative from the calculator’s algebra system. Domain, range and asymptotes are stated for real x.

What the graph of 1/(1 + e⁻ˣ) looks like

For large negative x the denominator is huge and the curve is close to 0, growing roughly like eˣ. For large positive x, e⁻ˣ is tiny and the curve is close to 1. The switch happens around x = 0, where the curve passes through ½ with slope ¼.

The general logistic growth curve is L / (1 + e^(−k(t − t₀))): L is the carrying capacity, k the growth rate and t₀ the midpoint, where growth is fastest.

The inverse of the sigmoid is the logit, ln(p/(1 − p)), which turns probabilities into log-odds.

Points of interest

Found numerically to about twelve digits in the window −8 ≤ x ≤ 8, then shown in exact form where one exists.

x-intercepts (roots): None in this window
Local maxima: None in this window
Local minima: None in this window
Inflection points: (0, 1/2)

Table of values

A few points to plot, with exact values where they are simple.

Values of 1/(1 + e⁻ˣ)
x	y = 1/(1 + e⁻ˣ)
−4	0.0179862
−2	0.119203
−1	0.268941
0	1/2
1	0.731059
2	0.880797
4	0.982014

How to graph 1/(1 + e⁻ˣ) by hand

1
Draw the asymptotes y = 0 and y = 1.
2
Plot (0, ½), (2, 0.88), (−2, 0.12), (4, 0.98) and (−4, 0.02).
3
Draw an S-curve with its steepest point at (0, ½).

Transformations

How the graph moves when you change the formula. The original is 1/(1 + e⁻ˣ) in blue.

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y = 1/(1 + e^(-3x)) Larger growth rate: a sharper switch.
y = 1/(1 + e^(-(x - 3))) Midpoint moved to x = 3.
y = 2/(1 + e^(-x)) Carrying capacity 2.

Where it shows up

Population growth limited by resources.
The spread of a technology or an epidemic before saturation.
Logistic regression and neural networks, where it turns a score into a probability.