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The graph of y = 1/x has two branches, one positive and one negative, which approach both axes without touching them. Free IGCSE Maths notes and 8 FAQs.
Reciprocal and Trigonometric Graphs is part of Quadratic and Other Graphs in IGCSE Edexcel Higher Maths.
2 min
Reading time
20
Lessons in this topic
IGCSE Edexcel Higher
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Reviewed by Revise Right
Content review
The graph of y = 1/x has two branches, one positive and one negative, which approach both axes without touching them. Free IGCSE Maths notes and 8 FAQs.
Use this lesson to stay inside Quadratic and Other Graphs while connecting the detail back to the wider Topic 3: Sequences, Functions and Graphs topic.
Read the core explanations from Reciprocal and Trigonometric Graphs before testing yourself from memory.
These graphs matter because they behave differently from linear, quadratic and cubic families. Reciprocal graphs split into branches and trig graphs repeat in cycles.
Your job here is shape recognition, sensible sketching and reading the main features accurately.
Worked example: Reciprocal shape
Describe the graph of y = 1/x.
Answer: A two-branch graph approaching both axes as asymptotes.
Worked example: Reading a sine graph
Describe the key pattern of y = sin x for 0 to 360 degrees.
Answer: A repeating wave through (0,0), (90,1), (180,0), (270,-1), (360,0).
A reciprocal graph does not have to be centred on the axes. In a form such as y = 1/(x - 2) + 1, the asymptotes shift to x = 2 and y = 1, but the two-branch reciprocal structure stays the same.
If you sketch y = 1/x and it touches either axis, the sketch is wrong. The axes are asymptotes, not intercepts.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
A periodic graph is a graph whose pattern repeats regularly. Sine and cosine graphs are periodic: they repeat in cycles, so they should not be drawn stopping after one bump.
The graph of y = 1/x has two branches, one positive and one negative, which approach both axes without touching them. The axes x = 0 and y = 0 are asymptotes because x cannot be zero.
For y = sin x from 0 to 360 degrees, the graph passes through (0,0), rises to a maximum at (90,1), returns to (180,0), falls to a minimum at (270,-1) and ends the cycle at (360,0). The wave then repeats.
The graph of y = 1/x cannot touch the axes because 1/0 is undefined, so x cannot be zero. The axes are asymptotes, not intercepts, so a sketch that touches either axis is wrong.
For y = 1/(x - 2) + 1, the asymptotes shift to x = 2 and y = 1. The graph keeps the same two-branch reciprocal structure as y = 1/x.
For y = 1/x, x = 1, 2, -1, -2 give y = 1, 1/2, -1, -1/2. The graph has one positive and one negative branch, approaching both axes as asymptotes.
Sine graphs repeat in a regular wave, so they are periodic graphs. Over one cycle from 0 to 360 degrees, y = sin x passes through (0,0), (90,1), (180,0), (270,-1) and (360,0), then the pattern repeats.
sin 90 = 1, the maximum of the first cycle of the sine graph, and sin 270 = -1, the minimum. sin 0 = 0 and sin 360 = 0, so a cycle starts and ends on the axis.