One Relationship, Different Representations

A function describes a relationship between:

inputs and outputs.

The same function can be represented in several different ways.

For example, imagine the rule:

Multiply the input by 2, then add 1.

We could represent this using an equation:

f(x) = 2x + 1

We could represent it using a table:

We could also represent it using a:

graph.

Although these representations look different, they all describe exactly the:

same mathematical relationship.

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The Main Representations of a Function

Functions can commonly be represented using:

• words
• tables
• equations
• graphs
• ordered pairs
• mapping diagrams

In this topic, we will concentrate mainly on:

tables, equations, and graphs.

Being able to move between these representations is an important part of:

understanding functions.

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Why Use Different Representations?

Each representation helps us see something different.

A table shows:

specific input-output pairs.

An equation shows:

the mathematical rule.

A graph shows:

the visual pattern or shape of the relationship.

Words explain:

what the relationship means.

No single representation is always best.

Different representations help us answer:

different questions.

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Representing Functions with Tables

A function table lists input values and their corresponding:

outputs.

For example:

f(x) = 3x + 2

Choose some x-values and calculate f(x).

Each row represents one:

input-output pair.

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Creating a Table from an Equation

Suppose:

f(x) = 2x - 3

We want to create a table.

Choose:

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

Now substitute each value into the equation.

For x = -2:

f(-2) = 2(-2) - 3

f(-2) = -7

For x = -1:

f(-1) = 2(-1) - 3

f(-1) = -5

For x = 0:

f(0) = 2(0) - 3

f(0) = -3

For x = 1:

f(1) = 2(1) - 3

f(1) = -1

For x = 2:

f(2) = 2(2) - 3

f(2) = 1

So:

The table represents the same function as:

f(x) = 2x - 3.

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Tables Show Specific Values

One advantage of a table is that it gives us:

exact values.

If we want to know f(2), we can simply find:

x = 2

and read the corresponding:

output.

In the previous example:

f(2) = 1

Tables are especially useful when we want to examine:

individual input-output pairs.

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Limitations of Tables

A table usually shows only:

some values of a function.

Consider:

f(x) = 2x - 3

A table might show x-values from -2 to 2.

But the function may also be defined for:

x = 10

x = 3.5

x = -100

and many other values.

The table does not necessarily show the function's:

entire domain.

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Representing Functions with Equations

An equation provides a mathematical rule connecting:

inputs and outputs.

For example:

f(x) = 4x + 3

tells us:

multiply the input by 4 and add 3.

An equation allows us to calculate the output for:

any allowed input.

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Equations Are Compact

Suppose a function contains thousands of possible:

input-output pairs.

A table containing every pair might be impossible to:

write.

But an equation such as:

f(x) = 5x - 2

describes the entire relationship in:

one short statement.

This is one major advantage of:

equations.

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Equations Reveal Structure

Consider:

f(x) = 3x + 4

The equation immediately tells us that this is a:

linear function.

We can identify:

slope = 3

and:

y-intercept = 4.

The equation reveals mathematical information that may be less obvious from a:

table.

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Representing Functions with Graphs

A graph represents a function visually.

Each point on the graph has coordinates:

(x, y)

or:

(x, f(x)).

For example, if:

f(2) = 5

then the graph contains the point:

(2, 5).

A graph is created by plotting many input-output pairs.

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Creating a Graph from a Table

Suppose we have:

Convert each row into an ordered pair:

(-2, -3)

(-1, -1)

(0, 1)

(1, 3)

(2, 5)

Plot these points on a coordinate plane.

Because they come from the linear function:

f(x) = 2x + 1

the points lie on a:

straight line.

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The Connection Between Tables and Graphs

Every row of a function table corresponds to:

one point on the graph.

For example:

This connection is fundamental.

A table and graph are not separate pieces of mathematics.

They are two ways of displaying the:

same input-output pairs.

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Graphs Show Overall Behaviour

A major advantage of a graph is that it shows the:

overall pattern.

From a graph we can often quickly see:

• whether the function increases or decreases
• where it crosses the axes
• whether it is linear or curved
• maximum or minimum values
• where the function changes direction
• domain and range
• unusual features

Graphs are especially useful for understanding the:

shape and behaviour of a function.

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Limitations of Graphs

Graphs may not always provide:

exact values.

If a point appears near:

(4, 7)

it may be difficult to determine whether the actual output is:

7

7.1

or:

6.95.

The accuracy depends partly on the:

scale and quality of the graph.

For exact calculations, an equation or table may sometimes be:

better.

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From Equation to Table

Consider:

f(x) = x²

Choose several inputs:

x = -3, -2, -1, 0, 1, 2, 3

Calculate the outputs.

Notice the pattern.

Different inputs can produce the:

same output.

For example:

f(-2) = 4

and:

f(2) = 4.

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From Table to Graph

The previous table produces the points:

(-3, 9)

(-2, 4)

(-1, 1)

(0, 0)

(1, 1)

(2, 4)

(3, 9)

Plotting these points reveals a:

parabola.

The table contains the values.

The equation contains the rule.

The graph reveals the:

shape.

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From Table to Equation

Sometimes we are given a table and need to determine the:

function rule.

Consider:

Look at how y changes.

Each time x increases by:

1

y increases by:

3.

This suggests a linear function with:

slope = 3.

When:

x = 0

we have:

y = 4.

Therefore, the y-intercept is:

4.

The equation is:

f(x) = 3x + 4

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Using First Differences

For a linear function, we can examine the:

first differences.

Consider:

The outputs change by:

+4, +4, +4, +4

The constant first difference tells us the function is:

linear.

The slope is:

4.

Since:

f(0) = 2

the equation is:

f(x) = 4x + 2.

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When x Does Not Increase by 1

Be careful.

Consider:

The y-values increase by:

6

but the x-values increase by:

2.

Therefore:

slope = change in y / change in x

slope = 6 / 2

slope = 3

Since y = 1 when x = 0:

f(x) = 3x + 1

We must compare changes in:

both variables.

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From Graph to Table

Suppose a graph passes through:

(-2, -5)

(-1, -2)

(0, 1)

(1, 4)

(2, 7)

We can record these points in a table:

The graph has now been converted into a:

table.

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From Graph to Equation

Suppose a straight-line graph crosses the y-axis at:

2

and rises:

3 units

for every:

1 unit to the right.

Then:

slope = 3

and:

y-intercept = 2.

Using:

y = mx + b

we obtain:

y = 3x + 2

or:

f(x) = 3x + 2.

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Finding Slope from Two Points

Suppose a graph contains:

(1, 4)

and:

(3, 10).

Calculate the change in y:

10 - 4 = 6

Calculate the change in x:

3 - 1 = 2

Therefore:

slope = 6 / 2

slope = 3

If the graph also crosses the y-axis at 1, then:

f(x) = 3x + 1.

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From Words to an Equation

Functions can also begin with a:

verbal description.

Suppose:

A taxi charges $4 initially and $2 for every kilometre travelled.

Let:

d = distance travelled

and:

C(d) = total cost.

The starting cost is:

4.

The cost increases by:

2 for every kilometre.

Therefore:

C(d) = 2d + 4

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From Words to a Table

Using:

C(d) = 2d + 4

we can create:

The table shows specific examples of the relationship described in:

words.

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From Words to a Graph

We can plot:

(0, 4)

(1, 6)

(2, 8)

(3, 10)

and so on.

The resulting graph shows that the total cost:

increases at a constant rate.

Now the same relationship has four representations:

Words: $4 starting fee plus $2/km

Equation: C(d) = 2d + 4

Table: specific distances and costs

Graph: visual representation of cost against distance

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Each Representation Answers Different Questions

Consider the taxi example.

If we ask:

What is the exact cost for 5 km?

The table may answer this quickly:

$14.

If we ask:

What rule calculates the cost for any distance?

The equation is useful:

C(d) = 2d + 4.

If we ask:

How does cost change as distance increases?

The graph provides an immediate:

visual picture.

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Comparing the Representations

Each representation provides a different:

view of the same relationship.

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Representing a Nonlinear Function

Consider:

f(x) = x² + 1

The equation tells us the:

rule.

A table gives:

When plotted, the points create a:

curved parabola.

This demonstrates that multiple representations are useful for:

nonlinear functions as well as linear functions.

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Recognizing Linear Relationships from Tables

Consider:

The output increases by:

3

every time the input increases by:

1.

This constant rate of change suggests a:

linear relationship.

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Recognizing Linear Relationships from Graphs

A linear relationship produces a:

straight-line graph.

The constant slope shows that the output changes at a:

constant rate.

If the graph curves, the rate of change is generally:

not constant.

Therefore, graphs make it easy to distinguish between:

linear and nonlinear relationships.

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Recognizing Nonlinear Relationships from Tables

Consider:

The changes in y are:

+1, +3, +5, +7

The first differences are:

not constant.

Therefore, the relationship is:

not linear.

In fact:

y = x².

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Ordered Pairs as a Bridge

Ordered pairs provide a useful connection between:

tables and graphs.

A row such as:

becomes:

(3, 8).

That ordered pair becomes:

a point on the graph.

So we can think of the process as:

Table row → Ordered pair → Graph point

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Equations Generate Ordered Pairs

An equation can generate as many ordered pairs as we:

need.

For:

f(x) = 2x + 1

choose x = 4.

Then:

f(4) = 9

This creates the ordered pair:

(4, 9).

Choose x = 10.

Then:

f(10) = 21

This creates:

(10, 21).

Every allowed input creates a corresponding:

point on the function's graph.

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Graphs Represent More Than the Points We Plot

When drawing a continuous function such as:

f(x) = 2x + 1

we might calculate only five points.

But the function contains infinitely many points between:

those calculated points.

For example:

x = 0.5

gives:

f(0.5) = 2

So:

(0.5, 2)

also lies on the graph.

The table helps us construct the graph, but it does not contain:

every possible point.

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Discrete Functions

Not every function should be drawn as a:

continuous line.

Suppose:

C = 5n

represents the cost of buying n notebooks.

Possible values of n might be:

0, 1, 2, 3, 4, ...

Values such as:

2.6 notebooks

do not make sense.

The graph should therefore contain:

separate points.

This is a:

discrete function.

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Continuous Functions

Suppose:

d(t) = 60t

represents distance travelled at 60 km/h.

Time could be:

1 hour

1.5 hours

1.57 hours

or many other values.

The inputs can vary continuously.

Therefore, the graph can normally be drawn as a:

continuous line.

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Converting Between Representations

A useful skill is being able to move freely between:

words ↔ equation ↔ table ↔ graph

For example:

Words

A quantity begins at 3 and increases by 2 for each unit of x.

↓

Equation

f(x) = 2x + 3

↓

Table

↓

Graph

Plot:

(0, 3), (1, 5), (2, 7), (3, 9)

and connect appropriately.

All four representations describe:

the same relationship.

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Worked Example 1: Equation to Table

Given:

f(x) = -2x + 5

complete the table.

For x = -1:

f(-1) = -2(-1) + 5 = 7

For x = 0:

f(0) = 5

For x = 1:

f(1) = 3

For x = 2:

f(2) = 1

For x = 3:

f(3) = -1

Therefore:

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Worked Example 2: Table to Equation

Consider:

The y-values increase by:

4

when x increases by 1.

Therefore:

slope = 4.

When x = 0:

y = -2.

Therefore:

y-intercept = -2.

The equation is:

f(x) = 4x - 2.

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Worked Example 3: Graph Information to Equation

A straight-line graph passes through:

(0, 6)

and:

(2, 10).

Calculate the slope:

slope = (10 - 6) / (2 - 0)

slope = 4 / 2

slope = 2

The y-intercept is:

6.

Therefore:

f(x) = 2x + 6.

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Worked Example 4: Words to Multiple Representations

A streaming service charges:

$8 per month plus a one-time $12 registration fee.

Let:

m = number of months

and:

C(m) = total cost.

Equation:

C(m) = 8m + 12

Table:

Graph:

Plot the points:

(0, 12)

(1, 20)

(2, 28)

(3, 36)

(4, 44)

Because months are normally counted in whole numbers for this model, the context may make the representation:

discrete.

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Checking That Representations Match

Suppose an equation is:

f(x) = 3x - 1

but a table contains:

Does the table represent the equation?

Check x = 2:

f(2) = 3(2) - 1

f(2) = 5

But the table says:

6.

Therefore, the table does:

not match the equation.

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Checking a Point Against an Equation

Suppose the graph of:

f(x) = 2x + 3

appears to contain the point:

(4, 11).

Check:

f(4) = 2(4) + 3

f(4) = 11

Therefore:

(4, 11)

does lie on the graph.

This is a useful way to verify whether different representations:

agree.

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Comparing Two Functions

Multiple representations also help us compare:

different functions.

Suppose:

f(x) = 2x + 1

and:

g(x) = x + 4.

A table can compare their outputs:

We can immediately see:

f(3) = g(3) = 7.

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Comparing Functions Graphically

If we graph:

f(x) = 2x + 1

and:

g(x) = x + 4

the lines intersect at:

(3, 7).

This shows visually that both functions produce the same output when:

x = 3.

The table and graph communicate the same:

intersection.

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Choosing the Best Representation

Different situations may favour different representations.

Use a table when:

• exact individual values matter
• the dataset is small
• you want to compare selected inputs and outputs

Use an equation when:

• you need the mathematical rule
• you want to calculate new values
• you want a compact representation

Use a graph when:

• you want to see the overall pattern
• you want to identify trends
• you want to compare functions visually
• you want to identify intersections or turning points

Use words when:

• the context and meaning need to be explained

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Representations Work Together

Strong mathematical understanding means being able to look at one representation and imagine:

the others.

When you see:

f(x) = 3x + 2

you should begin to imagine:

• a table increasing by 3
• a straight-line graph
• a y-intercept of 2
• an input-output rule
• a constant rate of change

These are not separate facts.

They are different features of:

one function.

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Multiple Representations in Science

Suppose an object moves at constant speed.

A scientist might describe the relationship in words:

The object moves at 5 m/s.

Equation:

d(t) = 5t

Table:

Graph:

A straight line through:

(0, 0)

with a slope of:

5.

Each representation provides useful information about the:

same physical motion.

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Multiple Representations in Business

Suppose a company charges:

$25 fixed fee plus $10 per item.

Words describe the:

pricing system.

Equation:

C = 10n + 25

A table can show costs for selected quantities.

A graph can show how cost changes as the number of items:

increases.

This is why functions are so useful for:

mathematical modelling.

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Multiple Representations in Everyday Life

The same idea appears in:

• phone plans
• taxi fares
• wages
• electricity costs
• distance and time
• temperature changes
• population growth
• savings
• scientific experiments
• business profits

In each situation, multiple representations help us understand the relationship from:

different perspectives.

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Common Mistake: Treating Representations as Separate

A table, equation, and graph are not:

different functions

simply because they look different.

If they represent the same input-output rule, they describe:

the same function.

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Common Mistake: Connecting Discrete Points

Suppose a graph represents:

number of concert tickets purchased.

Values such as:

2.4 tickets

do not make sense.

Therefore, the plotted points should not automatically be connected with:

a continuous line.

Always consider the:

context.

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Common Mistake: Reading a Graph Inaccurately

When converting a graph to a table:

• check the axis scale
• identify coordinates carefully
• include negative signs
• distinguish x from y
• do not estimate when an exact value is clearly shown

Graph reading requires:

precision.

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Common Mistake: Assuming Every Pattern Is Linear

A table may contain a clear pattern without being:

linear.

For example:

1, 4, 9, 16, 25

has a strong pattern.

But the differences are:

3, 5, 7, 9.

They are not constant.

The relationship is:

quadratic, not linear.

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The REPRESENT Strategy

When working with multiple representations, ask:

R — Relationship

What relationship is being described?

E — Equation

Can I write a mathematical rule?

P — Points

What input-output pairs are known?

R — Rate or Pattern

How do the values change?

E — Examine the Graph

What shape should the relationship have?

S — Same Relationship?

Do all representations agree?

E — Explain

What does the relationship mean?

N — Note Restrictions

Is the function discrete, continuous, or otherwise restricted?

T — Test

Choose an input and check that every representation gives the same output.

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Check Your Understanding

1. Name three common ways of representing a function.

2. What information does a table show particularly well?

3. What information does an equation show particularly well?

4. What information does a graph show particularly well?

5. Explain how a row in a table corresponds to a point on a graph.

6. Create a table for f(x) = 2x + 3 using x = 0, 1, 2, 3, 4.

7. Write the ordered pairs produced by your table.

8. Create a table for g(x) = x² - 1 using x = -2, -1, 0, 1, 2.

9. A table has outputs 5, 8, 11, 14 as x increases by 1. What is the constant first difference?

10. A table contains (0, 4), (1, 6), (2, 8), and (3, 10). Determine the equation.

11. A straight line has slope 5 and y-intercept -2. Write its equation.

12. Explain how you can convert an equation into a table.

13. Explain how you can convert a table into a graph.

14. Explain how you can convert a linear graph into an equation.

15. What does a constant first difference tell you about a function?

16. Why might a table fail to show the entire function?

17. Give one advantage and one limitation of a graph.

18. Give one advantage and one limitation of an equation.

19. Give one advantage and one limitation of a table.

20. Explain the difference between a discrete and continuous graph.

21. A taxi charges $6 initially plus $3 per kilometre. Write a function for the total cost.

22. Create a table for the taxi function for distances of 0, 1, 2, 3, and 4 km.

23. Explain what the slope of the taxi graph represents.

24. Explain what the y-intercept of the taxi graph represents.

25. Explain how a table, equation, graph, and verbal description can all represent the same function.

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Key Terms

• Function: Relationship assigning each allowed input exactly one output.
• Representation: A way of displaying or describing mathematical information.
• Table: Arrangement showing selected input-output pairs.
• Equation: Mathematical statement describing the rule connecting variables.
• Graph: Visual representation of the relationship between variables.
• Ordered pair: Pair of coordinates written as (x, y).
• Input: Value supplied to a function.
• Output: Value produced by a function.
• Function notation: Notation such as f(x) used to describe a function's output.
• Linear function: Function with a constant rate of change and a straight-line graph.
• Nonlinear function: Function whose graph is not a straight line.
• Slope: Rate of change of a linear function.
• Y-intercept: Point where a graph crosses the y-axis.
• First difference: Difference between consecutive output values.
• Discrete function: Function represented by separate input values.
• Continuous function: Function that can take all allowed values throughout intervals.
• Mathematical model: Mathematical representation of a real-world relationship.
• Rate of change: Amount by which one variable changes compared with another.
• Independent variable: Input variable.
• Dependent variable: Output variable.

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Key Takeaways

• A function can be represented using words, tables, equations, graphs, ordered pairs, and mapping diagrams.
• Different representations can describe exactly the same mathematical relationship.
• A table shows specific input-output pairs clearly.
• An equation gives a compact mathematical rule for the function.
• A graph shows the overall shape and behaviour of a function.
• Words help explain the meaning and context of a relationship.
• Every row of a function table corresponds to an ordered pair.
• Every ordered pair corresponds to a point on the graph.
• An equation can be converted into a table by choosing inputs and calculating outputs.
• A table can be converted into a graph by plotting its input-output pairs.
• A linear table can often be converted into an equation by identifying the slope and y-intercept.
• Constant first differences suggest a linear relationship when the x-values increase by equal amounts.
• If x-values do not increase by 1, calculate slope using change in y / change in x.
• Straight-line graphs represent linear relationships.
• Curved graphs generally represent nonlinear relationships.
• Tables usually show only selected values rather than every possible value of a function.
• Graphs reveal patterns clearly but may not always provide exact values.
• Equations allow outputs to be calculated for any allowed input.
• Discrete functions should not automatically be drawn as continuous lines.
• Context helps determine whether a function should be discrete or continuous.
• Different representations are useful for answering different types of questions.
• Checking the same input in several representations can verify that they describe the same function.
• Strong mathematical understanding means being able to move confidently between tables, equations, graphs, and words.
• Multiple representations provide different views of one underlying relationship, helping us understand functions more completely.