Trigonometric & exponential graphs – GCSE Maths

Higher · Grade 7 · Non-calculator

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Know the standard shapes: $y=\sin x$ and $y=\cos x$ oscillate between $-1$ and $1$ with period $360^{\circ }$; $y=\tan x$ has period $180^{\circ }$ and vertical asymptotes at $90^{\circ },270^{\circ },\dots$. $y=a^x$ (exponential growth for $a>1$) gets steep and never goes negative; $y=a^{-x}$ decays. $y=\sin x$ passes through $(0,0)$; $y=\cos x$ through $(0,1)$.

Transformations later will stretch these; here you sketch the parent, mark the key intercepts and turning points, and read solutions of $\sin x=0.5$ as the $x$-values where the graph is at height $0.5$ (two in $0^{\circ }$ to $360^{\circ }$ for sin).

$\sin$ and $\cos$ oscillate between $-1$ and $1$; $\tan$ repeats every $180^{\circ }$ and has asymptotes. $y=2^x$ always passes through $(0,1)$. CAST / the graph gives the second solution in $0^{\circ }$ to $360^{\circ }$.

Objectives

• Read sin and cos at the key angles.
• Read tan 45 and the period of sine.
• Read powers of 2 from the exponential graph.

Key points

• Sketch the sine and cosine curves from $0^{\circ }$ to $360^{\circ }$ with the key points marked.
• Sketch $y=\tan x$ with its asymptotes at $90^{\circ }$ and $270^{\circ }$.
• Use symmetry to find a second solution of $\sin x=a$ or $\cos x=a$ in a given range.
• Sketch $y=k^x$ for $k>1$, knowing it passes through $(0,1)$ and never touches the $x$-axis.

Worked examples

Questions and answers

GCSE Maths trigonometric & exponential graphs questions and answers. Try each one, then open the answer. Easy questions come first.

Common mistakes

Most lost marks on this topic are one slip.

1. Reading sine at 180 as 1

   The peak is at 90. At 180 the value is 0

2. Saying 2⁰ = 0

   A power of 0 is 1

3. Giving the sine period as 180

   The full repeat is 360 degrees

Exam tips

• Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
• If the question says “show that”, the result is given. The marks are for the steps that reach it.
• The method is the same on AQA-style and Edexcel-style papers.
• Always sketch the curve, even roughly, and mark the line $y=a$ — the sketch is your evidence for the second solution.
• Label the key points ($0$, $90$, $180$, $270$, $360$ on the $x$-axis; $1$ and $-1$ on the $y$-axis) on every trig sketch.

Printable worksheet

The same questions are in a PDF you can print. Answers start on a later page.

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More on Trigonometric & exponential graphs

• Trigonometric & exponential graphs worked examples and method
  The method in order, key facts and the full lesson explanation
• Trigonometric & exponential graphs practice questions
  A second set of questions with answers, plus quick checks

AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Last updated 9 November 2026.