Curve Sketching: Turning Points, Roots and Asymptotes

Sketching Without Plotting Every Point

Plotting a graph from a table of values is slow and can miss the important features: where the curve turns, where it crosses the axes, and what happens far away or near a forbidden value. Curve sketching is the skill of drawing the shape of a graph accurately from a few key facts. It is examined in Further Mathematics and in first-year calculus (MTH 101/102), and it is how engineers and scientists get a quick picture of how a quantity behaves. This tutorial gives a step-by-step checklist and works through a cubic, a quartic and two rational functions with asymptotes.

The Curve Sketching Checklist

Domain: which x-values are allowed? Look for division by zero, square roots of negatives and logarithms of non-positive numbers.
Intercepts: put x = 0 to find the y-intercept; solve y = 0 for the x-intercepts (roots).
Symmetry: if f(−x) = f(x) the graph is symmetric about the y-axis (an even function); if f(−x) = −f(x) it has rotational symmetry about the origin (an odd function).
Stationary points: solve dy/dx = 0 and classify each point using the second derivative or the sign of dy/dx.
Asymptotes: vertical asymptotes where the function blows up; horizontal or oblique asymptotes as x → ±∞.
Behaviour at the ends: what does y do as x becomes very large positive or negative?
Sketch: mark everything on the axes and join it up smoothly, consistent with all the facts.

Polynomial Curves: End Behaviour and Roots

For a polynomial, the term with the highest power controls the ends of the graph.

A cubic with a positive x³ coefficient goes from bottom left to top right; with a negative coefficient, from top left to bottom right.
A quartic with a positive x⁴ coefficient rises at both ends, like a W; with a negative one, it falls at both ends.

At a single root the curve crosses the axis. At a repeated (double) root such as (x − 2)², it touches the axis and turns back. At a triple root it crosses with a flattened S shape (a point of inflection).

Example 1: A Cubic Curve

Sketch y = x³ − 3x² − 9x + 27.

Factorise. Group the terms: x²(x − 3) − 9(x − 3) = (x − 3)(x² − 9) = (x − 3)²(x + 3).

Intercepts. y-intercept: 27. x-intercepts: x = −3 (crosses) and x = 3 (double root, touches).

Stationary points. dy/dx = 3x² − 6x − 9 = 3(x² − 2x − 3) = 3(x − 3)(x + 1), so x = 3 or x = −1.

At x = −1: y = −1 − 3 + 9 + 27 = 32. The second derivative is 6x − 6 = −12 < 0, so (−1, 32) is a maximum.
At x = 3: y = 27 − 27 − 27 + 27 = 0. The second derivative is 12 > 0, so (3, 0) is a minimum, sitting on the axis at the double root.

Ends. The x³ coefficient is positive, so y → −∞ as x → −∞ and y → +∞ as x → +∞.

Sketch. Take the points in order of x. From the bottom left, the curve crosses the axis at (−3, 0), climbs to the maximum at (−1, 32), falls through (0, 27) down to touch the axis at the minimum (3, 0), then rises again to the top right.

Example 2: A Quartic Curve with Symmetry

Sketch y = x⁴ − 4x².

Symmetry. f(−x) = (−x)⁴ − 4(−x)² = f(x), so the curve is symmetric about the y-axis.

Intercepts. y = x²(x² − 4) = x²(x − 2)(x + 2). The roots are x = 0 (double: touches) and x = ±2 (crosses). The y-intercept is 0.

Stationary points. dy/dx = 4x³ − 8x = 4x(x² − 2) = 0 gives x = 0 or x = ±√2.

At x = 0: y = 0. The second derivative 12x² − 8 = −8 < 0, so (0, 0) is a maximum.
At x = ±√2: y = 4 − 8 = −4. The second derivative is 24 − 8 = 16 > 0, so (±√2, −4) are minima.

Sketch. A W shape: down from the top left, through (−2, 0), to the minimum (−√2, −4), up to touch the origin, down to (√2, −4), then up through (2, 0) to the top right.

Asymptotes of Rational Functions

A rational function is a fraction of two polynomials, y = P(x)/Q(x).

Vertical asymptotes occur where Q(x) = 0 (and P(x) ≠ 0). Near them, y shoots off to +∞ or −∞.
Horizontal asymptotes describe y as x → ±∞. If the degree of P is less than the degree of Q, the asymptote is y = 0. If the degrees are equal, it is y = (leading coefficient of P) ÷ (leading coefficient of Q).
If the degree of P is exactly one more than that of Q, there is an oblique (slant) asymptote, found by polynomial division.

Example 3: y = (2x + 1)/(x − 1)

Domain. All x except x = 1.

Intercepts. x = 0 gives y = 1/(−1) = −1. y = 0 when 2x + 1 = 0, so x = −1/2.

Asymptotes. Vertical: x = 1. Horizontal: the degrees are equal, so y = 2/1 = 2.

Which side? Write y = 2 + 3/(x − 1) (since 2(x − 1) + 3 = 2x + 1). Just to the right of x = 1, 3/(x − 1) is large and positive, so y → +∞. Just to the left, y → −∞. For large positive x, y is slightly above 2; for large negative x, slightly below 2.

Stationary points. dy/dx = −3/(x − 1)², which is never zero, so there are none: the curve decreases on each side of the asymptote.

Sketch. Two branches, like a rectangular hyperbola centred at (1, 2): one in the top right region approaching both asymptotes, and one in the bottom left passing through (−1/2, 0) and (0, −1).

Example 4: An Oblique Asymptote

Sketch y = (x² + 3)/(x − 1).

Divide. x² + 3 = (x − 1)(x + 1) + 4, so y = x + 1 + 4/(x − 1). As x → ±∞, 4/(x − 1) → 0, so the curve approaches the oblique asymptote y = x + 1. The vertical asymptote is x = 1.

Intercepts. y-intercept: 3/(−1) = −3. x-intercepts: x² + 3 = 0 has no real solutions, so the curve never crosses the x-axis.

Stationary points. dy/dx = 1 − 4/(x − 1)² = 0 gives (x − 1)² = 4, so x − 1 = ±2, giving x = 3 or x = −1.

x = 3: y = (9 + 3)/2 = 6. This is a local minimum on the right-hand branch.
x = −1: y = (1 + 3)/(−2) = −2. This is a local maximum on the left-hand branch.

(Check with the second derivative 8/(x − 1)³: positive at x = 3, negative at x = −1.)

Sketch. The right branch comes down from +∞ near x = 1, reaches its minimum (3, 6) and then rises along y = x + 1. The left branch rises along y = x + 1 from the bottom left, reaches its maximum (−1, −2), passes through (0, −3) and plunges to −∞ as x approaches 1 from the left.

Common Mistakes in Curve Sketching

Letting a curve cross a vertical asymptote. It never can, because the function is undefined there. (A curve can cross a horizontal asymptote at finite x; the asymptote only describes the ends.)
Forgetting that a double root touches the axis rather than crossing it.
Drawing stationary points with the wrong nature: always classify them.
Labelling nothing. Examiners want intercepts, turning points and asymptote equations written on the sketch.

Summary

Work through domain, intercepts, symmetry, stationary points, asymptotes and end behaviour before drawing.
Polynomial ends follow the highest-power term; double roots touch the axis.
Rational functions have vertical asymptotes where the denominator is zero, and horizontal or oblique asymptotes found from the degrees or by division.
Rewriting y as "quotient + remainder/denominator" shows which side of an asymptote the curve lies on.