What are reciprocal graphs?

The reciprocal y = 1x has two separate branches, one in the top-right and one in the bottom-left for y = 1x. The curve never touches either axis: the x-axis and y-axis are asymptotes, lines the curve approaches but never meets. There is no value at x = 0.

State the equations of the two asymptotes of y = 1x. [2 marks]

SingaporeMathsSyllabus dot point

How do we recognise and sketch the standard non-linear graphs in the syllabus?

Recognise and sketch graphs of cubic, reciprocal and exponential functions, and describe their main features

A focused answer to the O-Level E-Maths outcome on standard graph shapes. Cubic, reciprocal and exponential curves, their asymptotes and symmetry, and recognising a function from its graph.

Generated by Claude Opus 4.88 min answerUpdated 2026-06-06

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
The answer
Examples in context
Try this

What this dot point is asking

SEAB wants you to recognise and sketch the standard non-linear graphs on the syllabus, the cubic, the reciprocal, and the exponential, and to describe their key features such as intercepts, symmetry and asymptotes. Knowing these shapes by heart lets you sketch quickly and interpret unfamiliar graphs.

The answer

Cubic graphs

The basic cubic $y = x^3$ passes through the origin, increases for all $x$ , and has rotational symmetry about the origin, falling steeply on the left and rising steeply on the right. Cubics of the form $y = ax^3$ keep this shape, steeper for larger $|a|$ and reflected when $a$ is negative.

Reciprocal graphs

The reciprocal $y = \dfrac{1}{x}$ has two separate branches, one in the top-right and one in the bottom-left for $y = \dfrac{1}{x}$ . The curve never touches either axis: the $x$ -axis and $y$ -axis are asymptotes, lines the curve approaches but never meets. There is no value at $x = 0$ .

Exponential graphs

The exponential $y = a^x$ (with $a > 1$ ) passes through $(0, 1)$ , since any base to the power zero is $1$ , and increases ever more steeply as $x$ grows. As $x$ becomes large and negative, the curve approaches the $x$ -axis from above without touching it, so $y = 0$ is an asymptote and $y$ is always positive.

Asymptotes and symmetry

An asymptote is a line the curve gets arbitrarily close to but never reaches. Recognising asymptotes (the axes for a reciprocal, the $x$ -axis for an exponential) and symmetry (origin symmetry for the basic cubic and reciprocal) helps you sketch accurately and identify a function from a given graph.

Worked example

Sketch $y = \dfrac{1}{x}$ for $x > 0$ and describe its behaviour at the extremes of this range.

Step 1: Calculate a few points

At $x = 0.5$ , $y = 2$ ; at $x = 1$ , $y = 1$ ; at $x = 2$ , $y = 0.5$ ; at $x = 4$ , $y = 0.25$ .

Step 2: Note the behaviour near zero

As $x$ approaches $0$ from the positive side, $y$ becomes very large; the curve rises steeply close to the $y$ -axis, which is a vertical asymptote.

Step 3: Note the behaviour for large x

As $x$ grows large, $y$ becomes very small and positive; the curve flattens towards the $x$ -axis, which is a horizontal asymptote.

Step 4: Sketch

Draw a smooth decreasing curve in the top-right region passing through $(1, 1)$ , rising steeply towards the $y$ -axis and flattening towards the $x$ -axis, touching neither.

Examples in context

Example 1. Population growth. Unchecked population or money under compound interest grows exponentially, modelled by $y = a^x$ . The ever-steepening curve captures why such growth seems slow at first then rapid, and it never falls below zero.

Example 2. Inverse relationships. Quantities in inverse proportion, such as the time for a journey against speed, trace a reciprocal curve. The asymptotes reflect that the time grows without limit as speed approaches zero and shrinks towards zero as speed grows large.

Try this

Q1. State the coordinates of the point where $y = 3^x$ crosses the $y$ -axis. [1 mark]

Cue. At $x = 0$ , $y = 3^0 = 1$ , the point $(0, 1)$ .

Q2. State the equations of the two asymptotes of $y = \dfrac{1}{x}$ . [2 marks]

Cue. The $x$ -axis $y = 0$ and the $y$ -axis $x = 0$ .

Q3. Describe one way the graph of $y = -x^3$ differs from $y = x^3$ . [1 mark]

Cue. It is the reflection in the $x$ -axis, falling from top-left to bottom-right instead of rising.

Exam-style practice questions

Practice questions written in the style of SEAB exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Original3 marksOn the same axes, describe the shape of the graphs of

y = x^3

and

y = \dfrac{1}{x}

for positive values of

x

, and state which one passes through the origin.

Show worked answer →

$y = x^3$ is a cubic that rises steeply, passing through the origin $(0, 0)$ , increasing for all $x$ with a flat point at the origin.

$y = \dfrac{1}{x}$ is a reciprocal curve that does not touch the origin; for positive $x$ it decreases from large values near the $y$ -axis towards zero as $x$ grows, with the axes as asymptotes.

Only $y = x^3$ passes through the origin.

Markers reward the rising cubic shape through the origin, the decreasing reciprocal curve approaching the axes, and identifying the cubic as the one through the origin.

Original3 marksThe graph of

y = 2^x

is an exponential curve. (a) State the value of

y

when

x = 0

. (b) Describe the behaviour of the curve as

x

becomes large and negative.