Plot and recognise quadratic, cubic, reciprocal and exponential graphs, read roots and turning points, use graphs to solve equations, and recognise the equation of a circle and real-life graphs (Higher tier for non-linear).

What this dot point is asking

Beyond straight lines, the CCEA Algebra strand expects you to recognise, plot and interpret a family of curves. You must plot and recognise quadratic, cubic, reciprocal and (at Higher tier) exponential graphs, read off roots and turning points, use graphs to solve equations, recognise the equation of a circle, and interpret real-life graphs such as distance-time and rate graphs. Recognising a graph's shape from its equation, and reading information back from a graph, are the central skills.

Quadratic graphs

A quadratic equation plots as a parabola, a symmetrical U-shape. If the $x^2$ coefficient is positive the parabola opens upwards with a minimum; if negative it opens downwards with a maximum. The graph is made from a table of values, plotting the points and joining them with a smooth curve, never straight segments.

The key features are the roots (where the curve crosses the $x$-axis, which are the solutions of $y = 0$), the $y$-intercept (where $x = 0$, the constant term), and the turning point (the lowest or highest point, found from completing the square or symmetry).

Cubic, reciprocal and exponential graphs

A cubic $y = ax^3 + \ldots$ has a characteristic S-shape and can cross the $x$-axis up to three times. A reciprocal $y = \tfrac{k}{x}$ has two separate branches that approach but never touch the axes (asymptotes), in opposite quadrants. An exponential $y = a^x$ (with $a > 1$) passes through $(0, 1)$ and increases more and more steeply; it models growth such as compound interest and populations. Recognising these shapes from the equation is examined directly.

Solving equations from graphs

A graph turns an equation into a picture, so solutions can be read off where curves cross.

The equation of a circle (Higher)

A circle centred at the origin with radius $r$ has equation $x^2 + y^2 = r^2$. So $x^2 + y^2 = 25$ is a circle of radius $5$ centred at $(0, 0)$. You should be able to write down the centre and radius from the equation, and recognise this graph among the others. CCEA also links this to the line-and-circle simultaneous case.

Real-life graphs

Graphs model real situations: a distance-time graph has gradient equal to speed (a horizontal section means stationary), and a conversion graph turns one quantity into another by reading across. Interpreting the gradient as a rate of change, and reading values from the axes, are the AO2 skills examined here.

Why this matters

Recognising and interpreting graphs connects every part of algebra: roots link to quadratic equations, intersections link to simultaneous equations, and gradient links to rates of change in measures and science. CCEA tests both drawing curves accurately from a table and extracting meaning from a given graph, so practise the smooth-curve technique and the reading-off method together.