Calculator
Enter a function, choose two x-values, and compute the secant slope, average rate of change, and secant equation.
Formula used
A secant line connects two points on the same curve. Its slope equals the average rate of change between those points.
When x₂ moves closer to x₁, the secant slope approaches the tangent slope. That makes secant lines useful for numerical differentiation and rate-of-change studies.
How to use this calculator
- Type the function expression using the chosen variable name.
- Enter the independent variable label, usually x or t.
- Choose a dependent variable label, usually y.
- Enter two different x-values for the points on the curve.
- Adjust decimal places and plot range if needed.
- Click the calculate button to show the result above the form.
- Review the computed points, slope, secant equation, and graph.
- Use the CSV or PDF buttons to export the current result.
Example data table
This sample uses f(x) = x² + 3x - 1 with several point pairs.
| Function | x₁ | x₂ | f(x₁) | f(x₂) | Secant slope |
|---|---|---|---|---|---|
| x² + 3x - 1 | 0 | 1 | -1 | 3 | 4 |
| x² + 3x - 1 | 1 | 4 | 3 | 27 | 8 |
| x² + 3x - 1 | 2 | 5 | 9 | 39 | 10 |
| x² + 3x - 1 | -2 | 3 | -3 | 17 | 4 |
FAQs
1) What is a secant line?
A secant line passes through two distinct points on a curve. It measures average change across an interval instead of instant change at one exact point.
2) What does the secant slope represent?
The secant slope equals the average rate of change of the function between x₁ and x₂. It compares how much the output changes per unit change in the input.
3) Why must x₁ and x₂ be different?
If x₁ equals x₂, the denominator becomes zero. That makes the secant slope undefined, because you would no longer have two distinct points for the line.
4) Can this calculator estimate a tangent slope?
Yes. Choose x₂ very close to x₁. The secant slope then approaches the tangent slope, which gives a practical numerical estimate of the derivative.
5) Which function types can I enter?
You can enter polynomials, roots, logarithms, exponentials, and many trigonometric expressions. Use supported names like sin, cos, tan, sqrt, log, and exp.
6) Why should I use explicit multiplication?
Explicit multiplication avoids ambiguity during parsing. Write 3*x, 2*(x+1), or x*(x-4) instead of compressed forms like 3x or 2(x+1).
7) What does a negative secant slope mean?
A negative secant slope means the function decreases on average from x₁ to x₂. As the input rises, the output falls over that chosen interval.
8) Do the graph and exports use my current settings?
Yes. The graph, CSV file, and PDF report use the latest valid calculation, including the entered expression, chosen variables, decimal places, and plotted interval.