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Edexcel · GCSE Maths · 1MA1 · Foundation and Higher

M13 · Quadratic and other graphs

Revision notes, worked examples and methods for quadratic and other graphs.

Revision notes ready · Quizzes and videos coming soon.

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 fourThe parabola has roots minus two and two and minimum at zero minus four.-3-2-10123-4-20246xy
    Graph of y equals x squared minus four
    Enlarge diagram: 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 = 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 xSeparate branches lie in quadrants one and three and approach both axes without touching.-4-2024-4-2024xy
    Graph of y equals one over x
    Enlarge diagram: 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 xThe curve passes through zero one, one two and two four, and increases more rapidly as x rises.-3-2-1012302468xy
    Graph of y equals two to the power x
    Enlarge diagram: Graph of y equals two to the power x
  • For y = kx with 0 < k < 1, the graph shows exponential decay. For example, y = ()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 turnSine starts at zero, peaks at ninety degrees and reaches minus one at two hundred and seventy. Cosine starts and ends at one.090180270360-101angle (°)ysin x — navycos x — rustsin xcos x
    Sine and cosine over one full turn
    Enlarge diagram: 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 asymptotesSeparate branches of tangent have zeros at minus one hundred eighty, zero and one hundred eighty degrees. Dashed vertical asymptotes at minus ninety and ninety are excluded from the graph.-180-90090180-4-2024angle (°)y
    Tangent graph and its asymptotes
    Enlarge diagram: 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

Quiz coming soon

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.

Revision video

Video coming soon.