Common graphs

Description

Recognising the shapes of common graphs is an important skill at GCSE. Each family of function produces a characteristic curve.

Linear: $y = mx + c$ — a straight line with gradient $m$ and $y$-intercept $c$.

Quadratic: $y = ax^2 + bx + c$ — a parabola, U-shaped (positive $a$) or n-shaped (negative $a$), with a minimum or maximum.

Cubic: $y = ax^3 + \ldots$ — an S-shaped curve with one or two turning points.

Reciprocal: $y = \frac{k}{x}$ — a hyperbola with asymptotes at $x = 0$ and $y = 0$.

Exponential: $y = a^x$ — steeply increasing (or decreasing), always positive, asymptote at $y = 0$.

Circle: $x^2 + y^2 = r^2$ — centred at origin with radius $r$.

Trig graphs: $y = \sin x$, $y = \cos x$ (both wave-shaped, period $360°$), $y = \tan x$ (period $180°$, with asymptotes).

Common error: confusing $y = x^2$ (quadratic) with $y = 2^x$ (exponential) — they look similar for positive $x$ but behave very differently.

Links

Equation of a straight line Equation of a circle