Quadratic, cubic and reciprocal graphs · GCSE Maths

Quadratic graphs

The graph of y = x² (or any quadratic) is a U-shaped curve called a parabola. If the x² term is negative, it is n-shaped.
The roots are where the curve crosses the x-axis. The y-intercept is where it crosses the y-axis. The turning point is the lowest (minimum) or highest (maximum) point.
The curve is symmetrical about a vertical line through the turning point.
y = x² − 4x + 3 has roots x = 1 and x = 3, y-intercept 3 and turning point (2, −1).

A cubic graph, such as y = x³, is S-shaped. If the x³ term is positive it goes from bottom left to top right.
The reciprocal graph y = 1/x has two separate curves (branches). They get closer and closer to the axes but never touch them, and x cannot be 0.
(Higher) An exponential graph y = kˣ (k > 1) passes through (0, 1), grows faster and faster, and never goes below the x-axis.
(Higher) y = sin x and y = cos x repeat every 360° and stay between −1 and 1. y = tan x repeats every 180°.

Complete a table of values, plot the points and join them with a smooth curve, not straight lines.
The solutions of x² − 4x + 3 = 0 are where y = x² − 4x + 3 crosses the x-axis. To solve x² − 4x + 3 = 2, find where the curve meets the line y = 2.

On a distance-time graph, the gradient is the speed. A horizontal line means the object has stopped.
On a conversion graph, read across from one axis and down to the other.

Parabola: The U-shaped or n-shaped curve of a quadratic graph.
Roots: The x-values where a graph crosses the x-axis.
Turning point: The minimum or maximum point of a curve.
Line of symmetry: A line that divides a graph into two mirror images.
Cubic graph: The S-shaped graph of an equation with x³ as the highest power.
Reciprocal graph: The graph of y = k/x, with two branches that never touch the axes.
Asymptote: A line that a curve gets closer and closer to but never touches.