Recognising and sketching standard graph shapes
You must recognise a small library of standard graphs by sight and sketch them quickly. Examiners give whole questions on shape recognition and matching equations to graphs.
Linear: y = mx + c
A straight line. Slope m, intercept c. Endless in both directions.
Quadratic: y = ax² + bx + c
A parabola: U-shape if a > 0, ∩-shape if a < 0. One turning point. Symmetric about a vertical line.
Cubic: y = x³ (and variants)
S-shape passing through the origin (for y = x³); rises from bottom-left to top-right.
General y = ax³ + bx² + cx + d can have 0 or 2 turning points.
Always passes from -∞ to +∞ (sign of a positive) — never bounded above or below.
Reciprocal: y = a/x (a > 0)
Two branches in opposite quadrants (Q1 and Q3 for positive a). The axes are asymptotes — the curve approaches but never touches them.
- As x → ∞, y → 0.
- As x → 0⁺, y → +∞.
For negative a (e.g. y = -2/x), branches sit in Q2 and Q4.
Exponential: y = a^x (a > 1)
Always positive (above the x-axis). Passes through (0, 1). Increases rapidly to the right; approaches y = 0 as x → -∞ (asymptote y = 0).
For 0 < a < 1 it falls instead of rising — a "decay" curve.
Square root: y = √x
Domain x ≥ 0; passes through (0, 0); rises slowly. Half a sideways parabola.
Trigonometric: y = sin x, y = cos x, y = tan x
- sin x: starts at (0, 0), rises to (90°, 1), back to (180°, 0), down to (270°, -1), back to (360°, 0). Wave amplitude 1, period 360°.
- cos x: same wave shifted left by 90°. Starts at (0, 1).
- tan x: rises from (0, 0); asymptotes at 90°, 270°, etc; period 180°.
✦Worked example— Worked example — match shape to equation
Given graphs A (parabola), B (S-shape rising), C (two reciprocal branches), D (exponential rise), match to:
y = x³, y = 2^x, y = -x² + 4, y = 6/x.
Answers: A ↔ y = -x² + 4 (downward parabola); B ↔ y = x³ (cubic); C ↔ y = 6/x (reciprocal); D ↔ y = 2^x (exponential).
⚠Common mistakes— Common mistakes (examiner traps)
- Confusing cubic with reciprocal. Cubic passes through the origin smoothly; reciprocal has a break (asymptote at x = 0).
- Drawing exponential through origin. It passes through (0, 1), not (0, 0).
- Drawing reciprocal branches in wrong quadrants. Sign of the coefficient determines which pair of opposite quadrants.
- Sketching trig with wrong period. sin and cos have period 360°; tan has period 180°.
- Forgetting asymptotes. When asked to sketch
y = 1/x, draw the asymptotes and indicate "→ 0" arrows.
➜Try this— Quick check
Sketch y = -x³. It is the reflection of y = x³ in the x-axis: starts top-left, passes through (0, 0), continues to bottom-right.
AI-generated · claude-opus-4-7 · v3-deep-algebra