How to Create a Variation Table from a Function

A variation table condenses the main behavior of a function into a small number of rows. It shows the domain, the sign of the first derivative, intervals of increase or decrease, extrema, limits, and discontinuities.

The standard process

1. Find the real domain of f.
2. Calculate f′(x).
3. Find critical numbers where f′(x)=0 or the derivative is undefined.
4. Split the domain at critical numbers and discontinuities.
5. Determine the sign of f′ on each interval.
6. Calculate function values and limits at important points.
7. Draw increasing and decreasing arrows in the f(x) row.

Worked example: f(x)=x^3-3x

The domain is all real numbers.

The derivative is f′(x)=3x^2-3=3(x-1)(x+1). Therefore the critical numbers are x=-1 and x=1.

The function values are f(-1)=2 and f(1)=-2. Thus (-1,2) is a local maximum and (1,-2) is a local minimum.

What appears in the table

The x-row contains -∞, -1, 1, and +∞. The derivative row shows +, 0, -, 0, +. The function row rises to 2, falls to -2, and rises again.

Discontinuities and excluded values

For a rational or logarithmic function, insert domain boundaries and vertical asymptotes. Use separate markers for excluded values and evaluate one-sided limits when the behavior differs on the two sides.

Create the table online

Open the Variation Table tool, choose a function, select the real domain or a custom interval, and calculate. You can then edit labels, line styles, arrows, exact values, and annotations before exporting.

Check the result

• Every interval should belong to the domain.
• The derivative sign should agree with the arrows.
• Extremum values should be evaluated in the original function.
• Infinity symbols should represent limits, not ordinary values.
• Manual edits should not replace mathematical verification.

Create the variation table for x^3-3x →

For a focused explanation of the derivative signs, read how to determine increasing and decreasing intervals.