Revision notes, worked examples and methods for quadratic and other graphs.
Notes and quizzes ready · 50 questions · Revision video ready.
Foundation shows shared notes and questions. Higher includes the shared content and labelled Higher extensions. Changing tier starts the notes and a new quiz set.
Revise the key ideas
Recognising graphs
The graph of y = x2 is a U-shaped parabola with minimum (0, 0), symmetric about the y-axis. y = −x2 opens downward. Generate values from the equation rather than guessing the curve. Graph of y equals x squared minus four
For y = x2 − 4, the y-intercept is −4 and the roots are x = −2 and x = 2. Roots are where y = 0, not where x = 0.
The turning point is the minimum or maximum. A quadratic's symmetry line passes through it; use symmetry to help read missing points.
The graph of y = x3 passes through the origin and increases from bottom left to top right. Its values are negative when x is negative.
The graph of y = 1x has two branches and is undefined at x = 0. The axes are asymptotes: the graph approaches them but never meets them. Graph of y equals one over x
Find roots from a factorisation when possible. y = (x − 1)(x + 3) has roots 1 and −3, so its symmetry line is halfway between them, x = −1.
Read an intersection's coordinates using the graph scale. State approximate values when the graph does not show an exact solution.
Higher — completing the square and exponentials
Write y = x2 + 6x + 5 as y = (x + 3)2 − 4. Its turning point is (−3, −4), because the squared term's smallest value is 0.
For y = 2x, values double when x increases by 1. It passes through (0, 1), remains positive and approaches y = 0 as x becomes very negative. Exponential growth is not constant addition. Graph of y equals two to the power x
For y = kx with 0 < k < 1, the graph shows exponential decay. For example, y = (12)x halves for each step to the right.
Higher — trigonometric graphs
Angles are in degrees at GCSE. y = sin x has period 360°, range −1 to 1, and zeros at multiples of 180°. y = cos x has the same period and range but starts at 1 when x = 0°. Sine and cosine over one full turn
y = tan x has period 180°, zeros at multiples of 180°, and vertical asymptotes at 90° + 180°n. It is undefined at those angles; do not join a curve across an asymptote. Tangent graph and its asymptotes
Use graph symmetry and periodicity to find more than one solution in a stated interval. sin x = 0.5 gives x = 30° and 150° for 0° ≤ x ≤ 360°.
Test yourself
50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.
Watch M13 · Quadratic and other graphs
Revise quadratic and other graphs with this narrated video. Use the player controls to pause, seek, adjust the volume or mute. Turn English captions on or off using the captions menu.
Mr Kidman's Classroom
