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Learn Index · Calculus Grade 11–12

Curve sketching

Using intercepts, critical points and concavity to sketch a graph.

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Putting it all together

Curve sketching combines algebra and calculus to draw an accurate graph without plotting dozens of points.

The checklist

Step Find Tool
1 Domain restrictions (denominators, roots, logs)
2 Intercepts set x = 0 and solve f(x) = 0
3 Symmetry even (f(−x) = f(x)) or odd (f(−x) = −f(x))
4 Asymptotes vertical, horizontal, oblique
5 Critical points; increasing/decreasing f′(x) and a sign chart
6 Local max and min first or second derivative test
7 Concavity and inflection points f″(x) and a sign chart
8 End behaviour limits as x → ±∞
9 Sketch combine everything

Example 1: a cubic

f(x) = x³ − 3x² − 9x + 5

  1. Domain: all real numbers.
  2. y-intercept: 5.
  3. No simple symmetry.
  4. No asymptotes (polynomial).
  5. f′(x) = 3x² − 6x − 9 = 3(x − 3)(x + 1) → x = −1, 3. Increasing on (−∞, −1) and (3, ∞), decreasing on (−1, 3).
  6. Local max f(−1) = −1 − 3 + 9 + 5 = 10; local min f(3) = 27 − 27 − 27 + 5 = −22.
  7. f″(x) = 6x − 6 = 0 → x = 1; inflection at (1, −6). Concave down for x < 1, up for x > 1.
  8. End behaviour: down on the left, up on the right.

Sketch: rises to (−1, 10), falls through (1, −6) to (3, −22), then rises.

Example 2: a rational function

f(x) = x/(x² − 4)

  1. Domain: x ≠ ±2.
  2. Intercept: (0, 0).
  3. Odd: f(−x) = −f(x) (symmetric about the origin).
  4. Vertical asymptotes x = ±2; horizontal asymptote y = 0.
  5. f′(x) = −(x² + 4)/(x² − 4)² < 0 → decreasing on each interval of the domain; no extrema.
  6. Use test values near the asymptotes to see which way the branches go.

Example 3: exponential

f(x) = xe^(−x)

  • f′(x) = e^(−x)(1 − x) → max at x = 1: f(1) = 1/e ≈ 0.37
  • f″(x) = e^(−x)(x − 2) → inflection at (2, 2/e²) ≈ (2, 0.27)
  • As x → ∞, f(x) → 0 (horizontal asymptote y = 0)

Tips

  • Organize results in a table before sketching.
  • Plot key points first (intercepts, extrema, inflection points).
  • Draw asymptotes as dashed lines.
  • Check your sketch with graphing technology.

Worked examples

Example 1: Inflection point of f(x) = x³ − 6x²? f″ = 6x − 12 → (2, −16).

Example 2: Is f(x) = x⁴ − 2x² even or odd? Even.

Example 3: Asymptotes of f(x) = 3/(x − 1)? x = 1, y = 0.

Common mistakes

  • Skipping the domain and missing asymptotes.
  • Sign chart errors.
  • Drawing smooth curves through asymptotes.

Try it yourself

For f(x) = x³ − 12x:

  1. Intercepts?
  2. Critical points and their type?
  3. Inflection point?
  4. Symmetry?

Answers: 1) (0, 0), (±2√3, 0) 2) local max (−2, 16), local min (2, −16) 3) (0, 0) 4) odd (origin symmetry)

Quick recap

  • Curve sketching uses domain, intercepts, asymptotes, f′ and f″.
  • f′ gives critical points and increasing/decreasing; f″ gives concavity and inflection points.
  • Check end behaviour and plot key points first.