Function families — shapes you must recognise

WJEC asks candidates to recognise and sketch six standard graph families. Memorise the shape, intercepts, and asymptotes.

Linear: y = mx + c

Straight line. Gradient m, y-intercept c. Crosses both axes at one point each (unless horizontal/vertical).

Quadratic: y = ax² + bx + c

Parabola. Opens upward if a > 0, downward if a < 0. Single turning point. May cross x-axis 0, 1 or 2 times.

Cubic: y = ax³ + ... (with a ≠ 0)

S-shape. May have two turning points (one max, one min) or a single point of inflection. Crosses x-axis 1, 2 or 3 times. y = x³ passes through (0, 0) with a horizontal tangent there.

Reciprocal: y = k/x

Two-branch hyperbola. Branches in opposite quadrants:

• k > 0: top-right and bottom-left.
• k < 0: top-left and bottom-right.

Asymptotes: x = 0 (vertical) and y = 0 (horizontal). Curve never touches the axes.

Exponential: y = aˣ (a > 0, a ≠ 1)

• a > 1 → grows steeply rightward; passes through (0, 1); approaches 0 as x → −∞.
• 0 < a < 1 → mirror image; decays rightward.

Always passes through (0, 1) since a⁰ = 1. Horizontal asymptote y = 0.

Trigonometric: y = sin x, y = cos x, y = tan x

For 0° ≤ x ≤ 360°:

• sin x: starts at 0, peaks +1 at 90°, returns to 0 at 180°, dips −1 at 270°, returns to 0 at 360°.
• cos x: starts at +1, falls through 0 at 90°, dips −1 at 180°, returns to 0 at 270°, peaks +1 at 360°.
• tan x: passes through 0, has vertical asymptotes at 90° and 270°. Period 180°.

Quick identification cues

• Two asymptotes? → reciprocal or tan.
• Passes through (0, 1) and approaches y = 0? → exponential.
• One turning point? → quadratic.
• Two turning points? → cubic.

WJEC exam tip

When asked to sketch (not plot), label intercepts, turning points, asymptotes and any roots that are given. A "sketch" earns marks for the right SHAPE plus the right LABELS — not for drawing accuracy.