Advanced Sigmoid Function Calculator

Calculator Inputs

Input x

Lower Asymptote A

Upper Asymptote B

Growth Rate k

Midpoint x₀

Graph Range Start

Graph Range End

Step Size

Use a negative growth rate for a decreasing sigmoid. Keep the upper asymptote above the lower asymptote.

Example Data Table

This example uses A = 0, B = 1, k = 1, and x₀ = 0.

x	Normalized Output p	Sigmoid y	Derivative
-4	0.017986	0.017986	0.017663
-2	0.119203	0.119203	0.104994
0	0.500000	0.500000	0.250000
2	0.880797	0.880797	0.104994
4	0.982014	0.982014	0.017663

Formula Used

General sigmoid form:
y = A + (B - A) / (1 + e^{-k(x - x₀)})

Normalized sigmoid:
p = 1 / (1 + e^{-k(x - x₀)})

Derivative:
dy/dx = (B - A) × k × p × (1 - p)

Logit:
logit(p) = ln(p / (1 - p))

The parameter A sets the lower bound. The parameter B sets the upper bound. The growth rate k controls steepness. The midpoint x₀ marks the inflection point, where the curve changes fastest.

How to Use This Calculator

Enter the x value you want to evaluate.
Set lower and upper asymptotes for the output range.
Choose the growth rate to control curve steepness.
Enter the midpoint where the sigmoid transitions fastest.
Provide graph start, end, and step values.
Click the calculate button to view results.
Review the output, derivative, odds, and logit values.
Use CSV or PDF buttons to export the generated results.

Frequently Asked Questions

1. What does this calculator compute?

It computes the sigmoid output, normalized value, derivative, odds, logit, midpoint output, and a full graph-ready curve table from your chosen parameters.

2. Can I use a negative growth rate?

Yes. A negative growth rate flips the curve, creating a decreasing sigmoid. The midpoint still marks the fastest rate of change.

3. What do the asymptotes mean?

The lower asymptote is the smallest long-run output. The upper asymptote is the largest long-run output. Together, they define the output range.

4. Why is the midpoint important?

The midpoint is where the sigmoid crosses its central level. It is also the inflection point, where the curve changes most rapidly.

5. What does the derivative tell me?

The derivative shows local sensitivity. Larger derivative values mean the output changes quickly for small changes in x near that location.

6. What is a logit value?

The logit transforms the normalized sigmoid output into log-odds. It is useful in classification models, statistics, and inverse mapping.

7. How should I choose the step size?

Use smaller steps for smoother plots and denser tables. Use larger steps for faster calculations over wide graph ranges.

8. Can I export the generated results?

Yes. After calculation, you can download the generated curve data as CSV and save a formatted summary as PDF.