Calculator Inputs
Example data table
Example parameters: L = 100, k = 0.8, x0 = 5, b = 0.
| x | y | First derivative |
|---|---|---|
| 0.0000 | 1.7986 | 1.4130 |
| 2.0000 | 8.3173 | 6.1004 |
| 4.0000 | 31.0026 | 17.1128 |
| 5.0000 | 50.0000 | 20.0000 |
| 6.0000 | 68.9974 | 17.1128 |
| 8.0000 | 91.6827 | 6.1004 |
| 10.0000 | 98.2014 | 1.4130 |
Formula used
Main logistic function
y = b + L / (1 + e^(-k(x - x0)))
Here, L is capacity, k is growth rate, x0 is midpoint, and b is the vertical shift.
First derivative
y' = k(y - b)(1 - (y - b)/L)
This measures slope. The largest slope occurs at x = x0.
Second derivative
y'' = k²(y - b)(1 - (y - b)/L)(1 - 2(y - b)/L)
This shows concavity and identifies the inflection point.
Inverse logistic solution
x = x0 - ln((1/p) - 1) / k where p = (y - b) / L
The inverse exists only when the target y stays strictly between the asymptotes.
How to use this calculator
- Enter the carrying capacity L. Keep it positive.
- Enter the growth rate k. Positive values rise, while negative values fall.
- Enter the midpoint x0, where the curve changes concavity.
- Set the vertical shift b if the lower baseline is not zero.
- Provide an input x to evaluate the function value and derivatives.
- Enter a target y if you also want the inverse solution for x.
- Choose graph start, graph end, and step size for the sampled table and chart.
- Press the calculate button to view results, graph the curve, and export CSV or PDF.
FAQs
1) What does L mean in the logistic function?
L is the carrying capacity. It sets the total vertical span of the curve above the shift value b. In many growth models, it represents the saturation limit.
2) What happens when k is negative?
A negative k flips the usual increasing logistic curve into a decreasing one. The same midpoint and asymptotes remain relevant, but the curve moves downward as x increases.
3) Why is the midpoint x0 important?
x0 is the inflection point. At this x value, the curve reaches half of its capacity above b, and the slope magnitude is greatest.
4) Why can the inverse fail for some target y values?
The inverse works only when the target lies strictly between the two horizontal asymptotes. A target on or beyond those limits would require an infinite or undefined x value.
5) What does the first derivative tell me?
The first derivative shows how quickly the logistic value changes at a selected x. Large positive or negative values indicate steeper movement. Near the asymptotes, the derivative approaches zero.
6) What does the second derivative show?
The second derivative describes concavity. It tells you whether the curve bends upward or downward at a given x, and it becomes zero at the inflection point.
7) Can this model probabilities?
Yes. When b = 0 and L = 1, the logistic function maps values into the open interval from zero to one. That form is common in probability and classification models.
8) How should I choose the graph step size?
Use a smaller step for smoother curves and more detailed tables. Use a larger step for faster rendering. This calculator automatically limits overly dense samples for better performance.