Function Variation Table & Sign Chart Calculator with Steps

The calculator above builds a complete variation table (also called a sign chart or monotonicity table) for five common function types: cubic, rational linear, rational quadratic over linear, biquadratic and quadratic. Each solution follows five steps: find the domain, differentiate, solve y’ = 0 and locate discontinuities, build the table, then state the intervals of increase and decrease and the local extrema. Every root is kept exact, including fractions and square roots.

What is a variation table?

A variation table summarizes how a function y = f(x) behaves across its domain. It has three rows:

• The x row: key x-values in increasing order: −∞, the critical points where y’ = 0, the points where the function is undefined, and +∞.
• The y’ row (sign chart): the sign of the derivative on each interval (+ or −), a 0 at each critical point, and a double bar || where the function is undefined.
• The y row: an upward arrow ↗ where the function increases, a downward arrow ↘ where it decreases, with the local maximum and minimum values and the limits written at the ends of the arrows.

The First Derivative Test

Let f be differentiable on an interval I. The derivative tells us the monotonicity of the function:

The First Derivative Test classifies a critical point c where f'(c) = 0:

• If f’ changes from + to − at c, then f has a local maximum at c.
• If f’ changes from − to + at c, then f has a local minimum at c.
• If f’ does not change sign, c is not a local extremum. For example, y = x³ has y’ = 3x² ≥ 0, so it is increasing on ℝ with no extremum at x = 0.

How to build a sign chart

Between two consecutive key x-values the derivative keeps one sign, so a single test point per interval is enough. Two shortcuts speed up the work:

• Linear factor ax + b: to the right of its root the factor has the sign of a; to the left it has the opposite sign.
• Quadratic factor ax² + bx + c: if Δ > 0, it has the sign opposite to a between the roots and the sign of a outside them. If Δ < 0, it always has the sign of a. If Δ = 0, it has the sign of a except at the double root.

For rational functions the denominator of y’ is a square, which is positive on the domain, so the sign of y’ is the sign of its numerator.

Monotonicity and extrema in five steps

1. Find the domain D. Polynomials are defined on ℝ; rational functions exclude the zeros of the denominator.
2. Compute y’. Use the power rule for polynomials and the quotient rule for rational functions.
3. Solve y’ = 0 and find discontinuities. Keep only roots in D. Compute the limits at infinity and the one-sided limits at each excluded point.
4. Build the table. Place the key x-values in order, fill in the signs of y’, draw the arrows and write the values in the y row.
5. State the conclusion. Read off the intervals of increase and decrease and the local maxima and minima.

Notation tip: for a rational function, write “increasing on (−∞, 1) and on (1, +∞)”, not on the union (−∞, 1) ∪ (1, +∞).

4 worked examples

Enter the same coefficients in the calculator to check every step.

Example 1. Cubic function y = x³ − 3x² + 2

The domain is ℝ. The derivative is y’ = 3x² − 6x = 3x(x − 2), so y’ = 0 at x = 0 or x = 2. Then y(0) = 2 and y(2) = −2. The leading coefficient is positive, so y → −∞ as x → −∞ and y → +∞ as x → +∞.

Conclusion: the function is increasing on (−∞, 0) and (2, +∞) and decreasing on (0, 2). It has a local maximum at x = 0 with y_max = 2 and a local minimum at x = 2 with y_min = −2.

Example 2. Biquadratic function y = −x⁴ + 2x² + 1

The domain is ℝ. The derivative is y’ = −4x³ + 4x = −4x(x² − 1), so y’ = 0 at x = −1, x = 0 and x = 1. Then y(±1) = 2 and y(0) = 1. The leading coefficient is negative, so both ends go down to −∞.

Conclusion: the function is increasing on (−∞, −1) and (0, 1) and decreasing on (−1, 0) and (1, +∞). It has local maxima at x = ±1 with y_max = 2 and a local minimum at x = 0 with y_min = 1.

Example 3. Rational linear function y = (2x + 1)/(x − 1)

The domain is ℝ \ {1}. The derivative is y’ = −3/(x − 1)² < 0 for every x ≠ 1, so y’ = 0 has no solution. Limits: lim y = 2 as x → ±∞, lim y = −∞ as x → 1⁻, lim y = +∞ as x → 1⁺.

Conclusion: the function is decreasing on (−∞, 1) and on (1, +∞) and has no local extrema. The graph has the vertical asymptote x = 1 and the horizontal asymptote y = 2.

Example 4. Rational function y = (x² − x + 1)/(x − 1)

The domain is ℝ \ {1}. By the quotient rule:

y’ = 0 at x = 0 or x = 2, both in the domain. Then y(0) = −1 and y(2) = 3. The numerator equals 1 > 0 at x = 1, so y → −∞ as x → 1⁻ and y → +∞ as x → 1⁺. At the ends, y → −∞ as x → −∞ and y → +∞ as x → +∞.

Conclusion: the function is increasing on (−∞, 0) and (2, +∞) and decreasing on (0, 1) and (1, 2). It has a local maximum at x = 0 with y_max = −1 and a local minimum at x = 2 with y_min = 3. The local maximum value is smaller than the local minimum value. That is fine: extrema are local, and the two points lie on different branches of the graph. The oblique asymptote is y = x.

Common mistakes

1. Leaving out an excluded x-value. For rational functions, the zero of the denominator must appear in the table, marked with ||.
2. Keeping a critical point outside the domain. A point not in D can never be an extremum.
3. Drawing the arrow the wrong way. A positive derivative means an upward arrow, even if the y-values are negative.
4. Calling a point an extremum when y’ does not change sign. A double root of y’, as for y = x³ at x = 0, is not an extremum.
5. Joining the intervals on both sides of a vertical asymptote. A rational linear function is monotonic on each interval separately, not on their union.

Frequently asked questions

What is a variation table in calculus?

A variation table is a three-row summary of a function. The x row lists the key x-values in increasing order, the y prime row shows the sign of the derivative on each interval, and the y row uses arrows to show where the function increases or decreases, together with local extreme values and limits.

How does the First Derivative Test find local extrema?

Find the critical points where the derivative equals 0. If the derivative changes from positive to negative at a critical point, the function has a local maximum there. If it changes from negative to positive, the function has a local minimum. If the sign does not change, there is no extremum.

What is the difference between a sign chart and a variation table?

A sign chart shows only the sign of the derivative on each interval. A variation table adds a third row with arrows, the extreme values and the limits, so it also shows how the function values move. The calculator builds both in one table.

What does the double bar mean in a variation table?

The double bar marks an x-value outside the domain, usually a zero of the denominator. The one-sided limits are written on each side of the bar. When they are infinite, the line through that x-value is a vertical asymptote.

Can a rational linear function have a local maximum or minimum?

No. For y = (ax + b)/(cx + d) with ad − bc not equal to 0, the derivative is (ad − bc)/(cx + d) squared. It never equals 0 and keeps the sign of ad − bc, so the function is monotonic on each interval of its domain.