Curve Sketching and Curve Analysis

Curve sketching (curve analysis) uses the function and its first and second derivatives to find the domain, intercepts, symmetry, asymptotes, intervals of increase and decrease, extrema, concavity and inflection points, and then draws an accurate graph. In brief: for f(x) = x³ − 3x, the zeros are 0 and ±√3, f′(x) = 3(x − 1)(x + 1) gives a local maximum (−1, 2) and a local minimum (1, −2), and f″(x) = 6x gives an inflection point at (0, 0). The function is odd, so the graph is symmetric about the origin.

A systematic checklist replaces guessing and tables of values. Each piece of information narrows the shape of the graph: the first derivative tells where the graph rises and falls, the second derivative tells how it bends. This guide gives a checklist, explains the sign tests for f′ and f″, works through a polynomial and a rational function with a figure, and provides a summary table for the steps.

What is the checklist for curve sketching?

1. Domain and points of discontinuity.
2. Intercepts: f(0), and the zeros f(x) = 0.
3. Symmetry: even, odd or neither, and periodicity.
4. Asymptotes: vertical, horizontal and oblique, and end behaviour.
5. First derivative: critical points, intervals of increase and decrease, local extrema.
6. Second derivative: concavity and inflection points.
7. Table: collect the information, compute values at the key points.
8. Sketch: plot the key points and asymptotes and join them with a smooth curve with the correct shape.

What does the first derivative tell you?

If f′(x) > 0 on an interval, f is increasing there, and if f′(x) < 0, f is decreasing. Critical points are the points where f′(x) = 0 or f′ does not exist. A local maximum occurs where f′ changes from positive to negative, and a local minimum where it changes from negative to positive. A point where f′ = 0 without a sign change (such as x = 0 for x³) is not an extremum.

What does the second derivative tell you?

If f″(x) > 0, the graph is concave up (it bends upward like a cup and lies above its tangents). If f″(x) < 0, it is concave down. An inflection point is a point where the concavity changes, which requires f″ to change sign (f″ = 0 is necessary but not sufficient: x⁴ has f″(0) = 0 but no inflection). The second derivative can also classify critical points: f′(a) = 0 and f″(a) > 0 gives a local minimum, f″(a) < 0 gives a local maximum.

Worked example: a cubic polynomial

Analyse f(x) = x³ − 3x.

1. Domain: all real numbers.
2. Intercepts: f(0) = 0. Zeros: x(x² − 3) = 0, so x = 0, ±√3 ≈ ±1.73.
3. Symmetry: f(−x) = −f(x), odd, so the graph is symmetric about the origin.
4. End behaviour: f → ∞ as x → ∞ and f → −∞ as x → −∞. No asymptotes.
5. f′(x) = 3x² − 3 = 3(x − 1)(x + 1). Critical points x = ±1. f′ > 0 for x < −1 and x > 1 (increasing), f′ < 0 for −1 < x < 1 (decreasing). Local maximum at x = −1: f(−1) = 2. Local minimum at x = 1: f(1) = −2.
6. f″(x) = 6x. f″ < 0 for x < 0 (concave down), f″ > 0 for x > 0 (concave up). Inflection point at (0, 0), where the slope is f′(0) = −3.

Worked example: a rational function

Analyse f(x) = x/(x² + 1).

1. Domain: all real numbers (the denominator is never 0).
2. Intercepts: f(0) = 0, the only zero.
3. Symmetry: f(−x) = −f(x), odd.
4. Asymptotes: no vertical. The degree of the numerator is less than that of the denominator, so y = 0 is a horizontal asymptote at both ends.
5. f′(x) = [(x² + 1) − x · 2x]/(x² + 1)² = (1 − x²)/(x² + 1)². Critical points x = ±1. f′ < 0 for x < −1, f′ > 0 for −1 < x < 1, f′ < 0 for x > 1. Local minimum at x = −1: f(−1) = −1/2. Local maximum at x = 1: f(1) = 1/2.
6. f″(x) = 2x(x² − 3)/(x² + 1)³. Zeros: x = 0, ±√3. Sign: f″ < 0 for x < −√3; f″ > 0 for −√3 < x < 0; f″ < 0 for 0 < x < √3; f″ > 0 for x > √3. So three inflection points: x = −√3, 0, √3, with the values f(±√3) = ±√3/4 ≈ ±0.433.

The graph rises from the asymptote y = 0 on the left, goes down to the minimum −1/2 at x = −1, rises through the origin to the maximum 1/2 at x = 1 and then falls back towards 0 from above. The range is [−1/2, 1/2].

Example 3: with an asymptote

f(x) = x²/(x − 1). Domain x ≠ 1; vertical asymptote x = 1; since the numerator has degree one more, divide: f(x) = x + 1 + 1/(x − 1), so the oblique asymptote is y = x + 1. f′(x) = (x² − 2x)/(x − 1)² = x(x − 2)/(x − 1)², critical points 0 and 2. Local maximum at (0, 0) and local minimum at (2, 4). The sign of f″ = 2/(x − 1)³: concave down for x < 1 and concave up for x > 1, with no inflection point, since x = 1 is not in the domain.

Summary table of the analysis

Where does curve analysis come from?

Newton classified the cubic curves in the 1700s and analysed their shapes, using his method of fluxions to find tangents, maxima and inflection points. The systematic use of the first and second derivatives for sketching became a standard part of calculus textbooks in the eighteenth and nineteenth centuries, with Euler’s Introductio and L’Hôpital’s work. Graphing calculators have made the process quick, but the analysis is still the way to make sure that the graph shows all features.

Where is it used?

Engineers analyse stress-strain curves, economists study cost and revenue curves (where the inflection point of the cost function marks the point of diminishing returns), and scientists look at the shape of growth curves. The method of checking the behaviour at all key points also underlies the detection of errors in numerical plots: a plot that misses a narrow peak can be corrected with the information from the derivative.

Which mistakes are common?

Assuming f″ = 0 means an inflection point

The sign of f″ must change. For f = x⁴, f″(0) = 0, but the graph is concave up on both sides.

Forgetting critical points where f′ does not exist

For f = |x| or f = x^2/3, the point x = 0 is critical though f′(0) does not exist.

Confusing increasing with concave up

A function can be increasing and concave down, like √x. The first derivative controls the direction, the second the bending.

Ignoring the domain and asymptotes

Do not draw the graph through a vertical asymptote. Mark the asymptotes first and fit the curve between them.