Standard graph shapes
Linear: y = mx + c
A straight line. Gradient m, y-intercept c.
Quadratic: y = ax² + bx + c
A parabola. If a > 0, opens upward (smile). If a < 0, opens downward (frown).
- Vertex (turning point): minimum if a > 0, maximum if a < 0.
- Roots: where y = 0, i.e. solutions of the quadratic.
- Axis of symmetry: x = −b / (2a).
Cubic: y = ax³ + …
General S-shape with up to two turning points. y = x³ has only an inflection at origin.
Reciprocal: y = k/x
Two branches (a hyperbola). Asymptotes at x = 0 and y = 0.
- k > 0: branches in 1st and 3rd quadrants.
- k < 0: branches in 2nd and 4th quadrants.
Exponential: y = aⁿ (n is a constant > 0)
- a > 1: exponential growth, passes through (0, 1), rises rapidly, asymptote y = 0.
- 0 < a < 1: exponential decay, passes through (0, 1), falls rapidly, asymptote y = 0.
Trigonometric (Higher tier)
| Function | Period | Range | Notable points |
|---|---|---|---|
| y = sin x | 360° | [−1, 1] | (0, 0), (90, 1), (180, 0), (270, −1), (360, 0) |
| y = cos x | 360° | [−1, 1] | (0, 1), (90, 0), (180, −1), (270, 0), (360, 1) |
| y = tan x | 180° | All real | Asymptotes at x = 90°, 270°… |
Recognising graph shapes from equations
CCEA frequently asks to match equations to sketches. Recognise the family first (quadratic, cubic, reciprocal, exponential, trig), then check the leading coefficient sign and the y-intercept.
Sketching y = x² + bx + c
- y-intercept: set x = 0 → y = c.
- Roots: factorise or use the quadratic formula.
- Vertex x-coordinate: x = −b/(2a).
- Substitute back to find vertex y-coordinate.
- Plot and draw smooth parabola.
Common CCEA exam tip
Always label key features: y-intercept, x-intercepts (roots), turning point, asymptotes (for reciprocals/exponentials). Each labelled feature is typically worth B1.
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