Graph sketching is tested from GCSE through university calculus. The techniques required at different levels overlap substantially — the checklist below applies at every level, with more items becoming available as the student progresses through the curriculum.
The nine-step checklist
1. Domain — What values of x is the function defined for? Mark excluded values (division by zero, square roots of negatives, logarithm arguments ≤ 0).
2. x-intercepts — Set y = 0, solve for x. These are where the graph crosses or touches the horizontal axis.
3. y-intercept — Set x = 0, compute f(0). One value (if 0 is in the domain).
4. Vertical asymptotes — Where does the denominator equal zero (and the numerator does not)? The function grows without bound near these values.
5. Horizontal/oblique asymptotes — Examine the limit as x → ±∞. For rational functions: compare degrees of numerator and denominator (horizontal asymptote is 0, the ratio of leading coefficients, or none, depending on the degree comparison; oblique when numerator degree exceeds denominator degree by exactly 1).
6. Critical points — Solve f'(x) = 0 and find where f' is undefined. These are candidates for local maxima and minima.
7. Increasing/decreasing intervals — Sign analysis of f'(x). Positive on an interval: function is increasing. Negative: decreasing.
8. Inflection points and concavity — Solve f''(x) = 0. Changes of sign in f'' are inflection points. f'' > 0: concave up. f'' < 0: concave down.
9. Sketch — Plot the intercepts, asymptotes, and critical/inflection points. Connect them with a curve consistent with the increasing/decreasing and concavity analysis.
Level 1 (GCSE/Grade 9): quadratics
f(x) = x² − 4x + 3
Domain: all real numbers. y-intercept: f(0) = 3. x-intercepts: x² − 4x + 3 = 0 → (x−1)(x−3) = 0 → x = 1 and x = 3. No asymptotes. Vertex (minimum): x = −b/(2a) = 2, f(2) = −1. Shape: opens upward (leading coefficient positive).
Level 2 (A-level/Pre-Calc): rational function
f(x) = (x+1)/(x−2)
Domain: x ≠ 2. y-intercept: f(0) = −1/2. x-intercept: x = −1. Vertical asymptote: x = 2. Horizontal asymptote: as x → ±∞, f(x) → 1 (same degree numerator and denominator, ratio of leading coefficients). No critical points (f'(x) = −3/(x−2)² < 0 everywhere — function is strictly decreasing on each piece).
Level 3 (Calculus): full curve analysis
f(x) = x³ − 3x² (classic cubic)
f'(x) = 3x² − 6x = 3x(x−2): critical points at x = 0 and x = 2. f(0) = 0 (local max), f(2) = −4 (local min). f''(x) = 6x − 6: inflection at x = 1, f(1) = −2. Concave down for x < 1, concave up for x > 1. Intercepts: f(0) = 0, f(3) = 0, f factor: x²(x−3) — so x = 0 (double) and x = 3.
Frequently Asked Questions
What is the difference between a critical point and an inflection point?
A critical point is where f'(x) = 0 or is undefined — a candidate for local extremum. An inflection point is where f''(x) changes sign — where the concavity of the curve changes direction. A critical point can be an inflection point (f'(x) = 0 and f''(x) = 0 and sign change in f''), but they are separate concepts.
Do I need to use calculus to sketch a graph?
At secondary (pre-calculus) level, use algebra: intercepts, symmetry, and end behaviour. At calculus level, derivatives provide the increasing/decreasing and concavity information. Both approaches are valid for the courses where they apply.