A LITTLE PRACTICE, EVERY DAY·GCSE MATHS
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Algebra · Higher

Exponential & trigonometric graphs

Exponential graphs have equation y = a·bˣ and grow (or decay) rapidly. Trigonometric graphs y = sin x, y = cos x and y = tan x have characteristic repeating shapes you must be able to sketch and read.

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Revising exponential & trigonometric graphs the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Exponential & trigonometric graphs GCSE” on the site for the complete guide.

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Key facts to remember

  • 1y = bˣ with b > 1: always positive, passes through (0, 1), grows rapidly to the right.
  • 2y = bˣ with 0 < b < 1: exponential decay.
  • 3Exponential graphs have a horizontal asymptote at y = 0.
  • 4y = sin x: wave between −1 and 1, period 360°, through origin.
  • 5y = cos x: wave between −1 and 1, period 360°, starts at (0, 1).
  • 6y = tan x: period 180°, asymptotes at 90°, 270°, …
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Worked examples

Example 1

Sketch y = 2ˣ for −2 ≤ x ≤ 3 and state the y-intercept.

Working

  1. x = −2 → 0.25; x = 0 → 1; x = 1 → 2; x = 2 → 4; x = 3 → 8.
  2. y-intercept at (0, 1).
  3. Smooth curve rising, horizontal asymptote y = 0 on the left.
AnswerExponential curve through (0,1), rising to (3, 8); y-intercept = 1.
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Common mistakes

✗Drawing the exponential curve crossing the x-axis — it never does.
✗Confusing sin and cos graphs.
✗Drawing tan as a continuous curve instead of separate branches.
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Exam tips

✓For y = a·bˣ always mark the y-intercept (0, a).
✓For trig graphs, mark the max, min and intercepts explicitly.
✓Know the first period (0° to 360°) for sin and cos by heart.

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