4.3PureCore

Reciprocal Graphs

Reciprocal functions like $y = \frac{1}{x}$ produce two separate curves that hug the axes but never touch them. The lines they approach are called asymptotes — a key idea for the rest of A-Level.

25 min Video by Zeeshan Zamurred Graphs and Transformations

Edexcel AS Level Maths: 4.3 Reciprocal GraphsWatch the full walkthrough before the notes below.

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What you'll be able to do

Sketch $y = \frac{k}{x}$ and $y = \frac{k}{x ^{2}}$
Identify horizontal and vertical asymptotes
Understand why the curve never meets the axes
See how the sign of k flips the graph

The shape of y = k/x

The graph of $y = \frac{k}{x}$ has two branches in opposite quadrants. For $k > 0$ they sit in the top-right and bottom-left; for $k < 0$ they swap to the other pair of quadrants.

y = \frac{k}{x}

Two branches; never touches either axis.

Asymptotes

An $asymptote$ is a line the curve gets ever closer to but never reaches. For $y = \frac{k}{x}$ the asymptotes are the $x$ -axis ( $y = 0$ ) and the $y$ -axis ( $x = 0$ ).

x = 0 and y = 0 are asymptotes

You can never divide by zero, and

\frac{k}{x}

is never exactly zero.

Tip — Dividing by zero is undefined → vertical asymptote. The output never reaches zero → horizontal asymptote.

y = k/x² and the sign of k

The graph of $y = \frac{k}{x ^{2}}$ has both branches on the $same$ side (both above the $x$ -axis when $k > 0$ ), because squaring removes the sign. Changing the sign of $k$ reflects the whole graph in the $x$ -axis.

Formula recap

y = \frac{k}{x}

Two branches in opposite quadrants.

y = \frac{k}{x ^{2}}

Both branches on the same side.

x = 0, y = 0

The asymptotes.

Common mistakes to avoid

Drawing the curve crossing or touching the axes.

Reciprocal curves approach the axes but never meet them.

Treating y = k/x and y = k/x² as the same shape.

The x² version has both branches on the same side of the x-axis.

Key takeaways

y = k/x has two branches in opposite quadrants.
The axes x = 0 and y = 0 are asymptotes — never touched.
y = k/x² keeps both branches on the same side.
The sign of k reflects the graph.

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