Graph recognition and sketching
OCR J560 expects students to recognise the standard "graph families" at a glance and to sketch each with key features labelled. Higher adds transformations and the unit-circle trig graphs.
Linear: y = mx + c
Straight line. Gradient m, y-intercept (0, c). Sketch with two clear points or with intercept and gradient triangle.
Quadratic: y = ax² + bx + c
Parabola. Opens up if a > 0, down if a < 0. Vertex at (−b/2a, ...). Symmetrical about x = −b/2a.
Cubic: y = ax³ + ... (e.g. y = x³, y = x³ − 3x)
Has up to two turning points (one local max + one local min) or none. Goes from −∞ to +∞ if a > 0; reverses for a < 0. y = x³ passes through the origin and has rotational symmetry about it.
Reciprocal: y = a/x
Two branches in opposite quadrants (Q1+Q3 if a > 0; Q2+Q4 if a < 0). Asymptotes: x-axis (y = 0) and y-axis (x = 0). Never crosses either axis.
Exponential: y = aᵇˣ (a > 0)
If b > 1: exponential growth — passes through (0, a), rises rapidly to the right, asymptote y = 0 to the left. If 0 < b < 1: decay — passes through (0, a), falls toward y = 0 to the right.
Trigonometric (Higher): y = sin x, y = cos x, y = tan x
| Function | Period | Range | Key features |
|---|---|---|---|
| sin x | 360° | [−1, 1] | sin 0 = 0, sin 90 = 1, sin 180 = 0, sin 270 = −1, sin 360 = 0 |
| cos x | 360° | [−1, 1] | cos 0 = 1, cos 90 = 0, cos 180 = −1, cos 270 = 0, cos 360 = 1 |
| tan x | 180° | all reals | tan 0 = 0, tan 45 = 1, asymptotes at x = 90°, 270° |
Sketch checklist (any function)
- Where does it cross the y-axis? (set x = 0)
- Where does it cross the x-axis? (set y = 0 if possible)
- Are there any asymptotes?
- What is the basic shape and direction?
- Are there turning points or symmetry?
OCR mark scheme conventions
- B1 for correct overall shape (curve type and direction).
- B1 for each labelled key feature (intercepts, asymptotes, turning points).
- "Sketch" means a freehand-quality but accurate rough drawing — proportions and key points right; perfect ruler precision not required.
⚠Common mistakes
- Drawing a quadratic as a "V" shape (that's |x|, not x²).
- Forgetting that y = a/x has TWO branches.
- Drawing tan x with the same period as sin/cos (it's 180°, not 360°).
- Putting an exponential through the origin (it doesn't — it crosses (0, a)).
AI-generated · claude-opus-4-7 · v3-ocr-maths-leaves