Representation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Representation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A mathematical representation is any format — diagram, equation, table, graph, or symbolic expression — used to encode and communicate a mathematical idea or relationship between quantities.

The same idea can be shown in multiple ways—each reveals different aspects.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A representation encodes a mathematical idea in a format — diagram, table, equation, or graph.

Common stuck point: The procedure for representation is the easy part; the trap is treating a graph, table, and equation of the same rule as different ideas. Asking "Am I encoding the same idea in a chosen format, knowing other formats encode it too?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I encoding the same idea in a chosen format, knowing other formats encode it too?

Worked Examples

Example 1

easy

Represent the set

A = \{1,2,3,4,5,6,7,8,9,10\}

with

B = \{2,4,6,8,10\}

and

C = \{1,3,5\}

using three different representations: (1) roster notation, (2) set-builder notation, (3) a Venn diagram description.

Answer

B = \{x \in A : 2 \mid x\},\quad C = \{x \in A : x \text{ odd},\; x \le 5\}

First step

Roster:

A=\{1,2,\ldots,10\}

B=\{2,4,6,8,10\}

C=\{1,3,5\}

Full solution

2
Set-builder: $B=\{x \in A : x \text{ is even}\}$ , $C=\{x \in A : x \text{ is odd and } x \le 5\}$ .
3
Venn diagram: Draw a rectangle for $A$ . Inside, draw circle $B$ (even numbers) and circle $C$ (odd $\le 5$ ). They do not overlap. Elements 7,9 are in $A$ but outside both circles.

The same mathematical object can be described in multiple ways. Switching representations often reveals different aspects of the structure and is a key problem-solving skill.

Example 2

medium

The inequality

|x - 3| < 2

can be represented as an absolute value inequality, a compound inequality, and an interval. Convert between all three.

Example 3

medium

Write the interval

-2 \le x < 5

in interval notation and as a set-builder set.

Example 4

medium

Represent the line through

(0,1)

with slope

2

in (a) slope-intercept form, (b) point-slope form.

Example 5

medium

Convert the recurring decimal

0.\overline{3}

to a fraction.

Example 6

medium

Represent the function 'add

4

then square' symbolically as

f(x)

Example 7

hard

Convert the point

(1, \sqrt{3})

from Cartesian to polar coordinates.

Example 8

hard

Express the parabola

y=x^2-4x+5

in vertex form.

Example 9

challenge

Convert the complex number

1+i

from rectangular to polar form.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Write the solution set of

x^2 = 9

in (a) roster notation, (b) set-builder notation.

Example 2

medium

A function

f

triples its input. Represent

f

as (a) a formula, (b) a table for inputs

\{1,2,3,4\}

, (c) a verbal description.

Example 3

easy

The same function is shown as a table, a graph, and the equation

y=2x

. These are three different what?

Example 4

easy

To see instantly where

f(x)=x^2-4

crosses the

x

-axis, which representation is most direct: a graph or a table of

100

values?

Example 5

easy

Is the graph of

f(x)

the same thing as the function

f

itself?

Example 6

easy

Which representation makes it easiest to compute

f(3)

for

f(x)=x^2+1

: the equation or a graph?

Example 7

easy

A pie chart and a bar chart show the same survey data. What does this illustrate about representation?

Example 8

easy

A table lists

f(0),f(1),f(2)

. Does it tell you the value of

f(1.5)

Example 9

easy

To show that two angles are equal, which representation usually helps most: a labeled diagram or a paragraph of words?

Example 10

easy

The number

0.5

, the fraction

\tfrac{1}{2}

, and '

50\%

' are three representations of what single value?

Example 11

medium

A motion problem is easier in a position-vs-time graph than in a paragraph. Name the representation switch and the feature it exposes.

Example 12

medium

Why might representing a recurring decimal

0.\overline{3}

as the fraction

\tfrac13

make a proof cleaner?

Example 13

medium

A system of two linear equations can be a pair of lines (graph) or a matrix equation

A\mathbf{x}=\mathbf{b}

. Which representation generalizes better to

100

equations, and why?

Example 14

medium

To prove a number is divisible by

9

, representing it via its digit sum helps. What feature does the digit-sum representation expose?

Example 15

medium

A probability problem is stated in words. Drawing a tree diagram changes the representation. What does the tree make explicit?

Example 16

medium

Complex number

z

can be written

a+bi

r(\cos\theta + i\sin\theta)

. Which representation makes multiplying two complex numbers simplest?

Example 17

medium

A data set's spread is asked for. Why might a box plot be a better representation than listing all

500

raw values?

Example 18

medium

Switching from

\sum

notation to writing out terms

1+2+\dots+n

is a representation change. When is the expanded form the BETTER representation?

Example 19

medium

To add the fractions

\tfrac13

and

\tfrac16

quickly, which representation helps: a common-denominator form or a decimal approximation?

Example 20

challenge

A graph appears to show

f(x)=\frac{x^2-1}{x-1}

as the line

y=x+1

. Explain how the graphical representation can mislead, and which representation corrects it.

Example 21

challenge

Counting lattice paths from

(0,0)

(m,n)

can be represented as binomial coefficients or as a grid-walk picture. Show both representations give

\binom{m+n}{m}

and explain why the symbolic one scales.

Example 22

challenge

An optimization is stated as 'maximize area of a rectangle with perimeter

20

.' Show how the algebraic representation

A=x(10-x)

reveals the maximum that the verbal representation hides.

Example 23

easy

Convert

\frac{3}{4}

to a decimal.

Example 24

easy

Convert

0.6

to a percent.

Example 25

medium

A function doubles its input and adds

1

. Write the rule symbolically.

Example 26

medium

Represent the equation

y = x^2

as a table for

x = -2, -1, 0, 1, 2

Example 27

easy

Which representation is best for spotting symmetry of

f(x)=x^2

— the equation or the graph?

Example 28

medium

Convert

\frac{5}{8}

to a percent.

Example 29

medium

Express the relation 'twice as old' between Ann's age

A

and Ben's age

B

symbolically.

Example 30

medium

Express

5 \times 10^3

in standard (positional) notation.

Example 31

medium

Express

0.00042

in scientific notation.

Example 32

medium

Roster form: list the elements of

\{x \in \mathbb{N} : x \le 5\}

Example 33

medium

Represent the line

y=3x-2

as a

y

-intercept and slope pair.

Example 34

medium

Convert

\tfrac{7}{20}

to a decimal.

Example 35

medium

Express '

x

is at most

3

' as an inequality.

Example 36

medium

Write the inequality

|x| \le 4

without absolute value.

Example 37

hard

The map

f

sends

1 \to a

2 \to b

3 \to c

. Write

f

as a set of ordered pairs.

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Related Concepts

Multiple Viewpoints Abstraction