GCSE Maths Graphs of functions revision

Linear graphs are straight lines y = mx + c. Quadratics y = x² + bx + c are U-shaped parabolas with one turning point. Cubics y = x³ + … swerve in an S: both ends run to infinity in opposite directions, and y = x³ passes through (0, 0) with a flat moment there. The reciprocal y = 1⁄x lives in two branches — top-right and bottom-left — and never touches either axis (x ≠ 0). Reading a family from its equation is the first skill: the highest power of x names the family.

Every family has tell-tale points. A quadratic y = ax² + bx + c crosses the y-axis at c. The cubic y = x³ passes through the origin with y negative for x negative. The reciprocal has no intercepts at all. In tables, a quadratic’s second differences are constant (Chapter 10), a cubic’s third differences are, and 1⁄x deals in products: (2, 0.5), (4, 0.25), (−1, −1).

Higher. y = kˣ always passes through (0, 1), because anything to the power 0 is 1. For k > 1 the curve climbs ever faster (growth: y = 2ˣ gives 1, 2, 4, 8, 16); for 0 < k < 1 it sinks towards the x-axis (decay: y = (½)ˣ gives 1, ½, ¼). The x-axis is an asymptote — the curve never touches it. These are exactly the compound-interest and depreciation shapes from Chapter 2.

The trigonometric graphs wave: y = sin x starts at 0, peaks at 1 when x = 90°, returns to 0 at 180°, dips to −1 at 270°. y = cos x starts at 1, crosses zero at 90°, bottoms at 180° — cos is sin shifted 90° left. Both repeat every 360° and sit between −1 and 1. y = tan x repeats every 180° and shoots off to infinity at 90°, 270°, … where the graph has vertical asymptotes.

Higher. Starting from y = f(x): y = f(x) + a slides the whole graph up by a; y = f(x) − a slides down. y = f(x + a) slides left by a — inside the bracket is the backwards one, so y = f(x − 3) slides right 3. y = −f(x) reflects in the x-axis (every y flips sign); y = f(−x) reflects in the y-axis.

Tracking points keeps it honest: if (2, 8) sits on y = x³, then y = x³ + 5 passes through (2, 13), y = (x + 1)³ passes through (1, 8) (the x dropped by 1), and y = −x³ passes through (2, −8). Translations never change the shape — only its position or facing.

A distance–time graph turns steepness into speed: a steep section is fast, a flat section is a stop, and a downhill section is a return journey. The gradient between two points is distance ÷ time. On curved graphs (Higher), the gradient at a point is estimated by drawing a tangent — a journey speeding up curves upwards, and the tangent at 3 minutes might have gradient 0.8, read as 0.8 km per minute at that instant.

Higher. The area under a velocity–time graph is the distance travelled. For straight sections, use trapezium areas: a graph at 4 m/s for 3 s then rising to 10 m/s over the next 6 s covers (4 + 10)⁄2 × 6 = 42 m on the second stage alone (plus 12 m on the first). Rough curves are divided into trapezium strips; more strips, better estimate.

Higher. A circle centred at the origin with radius r has equation x² + y² = r²: the square of the distance from the origin is r². So x² + y² = 25 is radius 5, and the point (3, 4) lies on it because 9 + 16 = 25. A radius from the origin to (a, b) has gradient b⁄a.

The tangent at a point is perpendicular to the radius there. At (3, 4) the radius has gradient 4⁄3, so the tangent has gradient −¾; through (3, 4), the tangent is y − 4 = −¾(x − 3), i.e. y = −¾x + 6¼. At the point (5, 0) the radius is horizontal, so the tangent is the vertical line x = 5 — a reminder that some tangents are not functions.