Connect representations of linear relations

Ontario Grade 9 Mathematics

Algebra

Moving Between Tables, Graphs, and Equations

A linear relation is a relationship between two quantities where one quantity changes at a constant rate as the other increases. You have already seen this idea in Grade 8 when you plotted points and noticed they lined up perfectly. In this lesson you will learn that every linear relation can be shown in three ways: a table of values, a graph, and an equation. These three representations carry exactly the same information — they are just different languages for the same relationship. Being able to move between them freely is a powerful skill. You will start with a real-world situation, build a table, draw a graph, and then write an equation, so that each step grows naturally from the one before it.

What you will learn

Identify the same linear relation presented as a table of values, a graph, and an equation.
Explain how the constant rate of change and starting value appear in each representation.
Move from a table of values to a graph and then to an equation.
Move from an equation to a table of values and a graph.
Decide which representation is most useful for answering a given question about a linear relation.

What Is a Linear Relation? A Quick Review

Before connecting representations, make sure you understand what makes a relation linear. A relation pairs values of one variable with values of another. Call the two variables

x

and

y

. When you move from one

x

-value to the next by adding 1, the matching

y

-value always changes by the same fixed amount. That fixed amount is called the constant rate of change.

Because the rate of change never varies, the points always fall on a perfectly straight line when graphed. That is exactly why the relation is called linear — it produces a line. If the differences in

y

are not all equal, the relation is not linear and you will not get a straight line.

Two numbers fully describe any linear relation: (1) the rate of change, which tells you how steeply

y

rises or falls as

x

increases, and (2) the initial value, which is the value of

y

when

x = 0

. You will see both of these numbers in every representation.

rate of change =

\frac{change in y}{change in x}

A linear relation has a constant rate of change.
Its graph is always a straight line.
Two numbers define it completely: the rate of change and the initial value.
The rate of change is found by calculating the difference in $y$ divided by the difference in $x$ between any two points.

The Table of Values Representation

A table of values lists paired numbers. The left column shows the

x

-values (the input) and the right column shows the matching

y

-values (the output). Here is a concrete situation to anchor the idea: a student earns CAD 12 for every hour of yard work, and the neighbour pays an extra CAD 5 up front for supplies. The student's total pay depends on the number of hours worked.

Let

x

represent the number of hours worked and

y

represent the total pay in dollars. When

x = 0

, the student has not worked yet but already received CAD 5, so

y = 5

. Each additional hour adds CAD 12. Working through a few hours gives the table shown below.

To check that a table represents a linear relation, subtract each

y

-value from the next one. If every difference is the same, the rate of change is constant. In this table, every step from one row to the next adds 1 to

x

and 12 to

y

, so the rate of change is

\frac{12}{1} = 12

. The initial value (when

x = 0

) is

5

. These two numbers are the key to the other two representations.

rate of change =

\frac{\Delta y}{\Delta x}

=

\frac{12}{1}

= 12

A table lists input–output pairs.
Check linearity by confirming equal differences in $y$ for equal steps in $x$ .
The rate of change equals the common difference in $y$ when $x$ increases by 1.
The initial value is the $y$ -value in the row where $x = 0$ .

The Graph Representation

To draw the graph, plot each ordered pair

(x, y)

from the table on a coordinate grid. The horizontal axis represents

x

(hours worked) and the vertical axis represents

y

(total pay in dollars). Plot the point

(0, 5)

, then

(1, 17)

, then

(2, 29)

, and so on. Once plotted, draw a straight line through all the points and extend it in both directions as far as the context allows.

Two features of the graph connect directly to the numbers you found in the table. First, the line crosses the vertical axis at

y = 5

. This crossing point is called the vertical intercept (also written as the

y

-intercept). It matches the initial value from the table. Second, for every 1 unit you move to the right along the horizontal axis, the line rises 12 units. This steepness is called the slope, and it matches the rate of change from the table.

The slope tells you the direction of the line as well as its steepness. A positive slope means the line goes up from left to right (the student earns more as they work more). A negative slope would mean the line goes down (the quantity decreases). A slope of zero would produce a perfectly horizontal line (no change in

y

at all).

\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}

Each row of the table becomes one plotted point $(x, y)$ .
The vertical intercept on the graph equals the initial value from the table.
The slope on the graph equals the rate of change from the table.
A positive slope rises left to right; a negative slope falls left to right.

The Equation Representation

An equation is a compact, precise way to describe the relation for every possible value of

x

, not just the ones listed in a table. For a linear relation, the equation has the form

y = mx + b

, where

m

is the slope (rate of change) and

b

is the vertical intercept (initial value). This is called the slope-intercept form of a linear equation.

Using the values from the yard-work situation: the rate of change is

12

and the initial value is

5

. Substituting gives

y = 12x + 5

. You can verify this equation using the table: when

x = 3

, the equation gives

y = 12(3) + 5 = 36 + 5 = 41

, which matches the table value.

Reading an equation works the same way in reverse. If someone hands you the equation

y = -3x + 20

, you immediately know the slope is

-3

(the

y

-value decreases by 3 each time

x

increases by 1) and the vertical intercept is

20

(the line crosses the

y

-axis at the point

(0, 20)

). You can then build a table by substituting

x

-values, and you can sketch the graph by plotting the intercept and counting the slope.

y = mx + b

The equation $y = mx + b$ represents every linear relation.
$m$ is the slope, equal to the rate of change.
$b$ is the vertical intercept, equal to the initial value.
Substitute any $x$ -value into the equation to find the matching $y$ -value without a table.

Choosing the Most Useful Representation

All three representations — table, graph, and equation — show the same relation. The best choice depends on what question you are trying to answer. A table is easiest when you need exact values for a small number of specific inputs, such as checking pay for exactly 3 hours. A graph is best when you want a visual overview, for example to see when two relations intersect or to spot trends at a glance. An equation is most powerful when you need to calculate outputs for many inputs quickly, or when

x

can be any number, not just whole numbers.

Being able to move between representations is the real skill. If you are given only a graph, you can read the vertical intercept and calculate the slope from two clear points, then write the equation. If you are given only an equation, you can substitute values to build a table and then plot the graph. If you are given only a table, you can calculate the rate of change and read the initial value to write the equation, and then graph it.

Practise translating in every direction: table → graph, table → equation, equation → table, equation → graph, graph → table, graph → equation. Each direction tests a slightly different understanding, but they all rest on the same two key numbers: slope and vertical intercept.

y = mx + b \quad (m = \text{slope},\; b = \text{vertical intercept})

Use a table for exact values at specific inputs.
Use a graph for visual trends and comparisons.
Use an equation for quick calculation at any input value.
All three representations contain the same slope and vertical intercept.

Yard-Work Pay: Table of Values

Hours worked ( $x$ )	Total pay in dollars ( $y$ )	Check: rate of change
0	5	— (starting point)
1	17	$17 - 5 = 12$ ✓
2	29	$29 - 17 = 12$ ✓
3	41	$41 - 29 = 12$ ✓
4	53	$53 - 41 = 12$ ✓

Three Representations at a Glance

Representation	Where to find the slope ( $m$ )	Where to find the vertical intercept ( $b$ )
Table of values	Constant difference in $y$ ÷ step in $x$	$y$ -value in the row where $x = 0$
Graph	Rise ÷ run between any two points on the line	Point where the line crosses the $y$ -axis
Equation $y = mx + b$	Coefficient of $x$ (the number multiplied by $x$ )	Constant term added at the end

Worked example

Moving Between All Three Representations

A phone plan charges a flat monthly fee of CAD 20 plus CAD 0.05 for each text message sent. Let

x

represent the number of text messages sent in a month and

y

represent the total monthly cost in dollars. (a) Build a table of values for

x = 0, 20, 40, 60, 80

. (b) Write an equation for this relation. (c) Identify the slope and vertical intercept, and explain what each means in this context. (d) Use the equation to find the cost when 150 messages are sent.

Step 1 — Identify the rate of change and initial value
The flat fee is CAD 20. This is paid regardless of how many texts are sent, so it is the initial value: when $x = 0$ , $y = 20$ . Each text costs CAD 0.05, so each time $x$ increases by 1, $y$ increases by 0.05. The rate of change is 0.05.
initial value = 20, rate of change = 0.05
Step 2 — Build the table of values
Substitute each given $x$ -value into the pattern: multiply the number of texts by 0.05 and add 20. For example, when $x = 40$ : $0.05 \times 40 = 2.00$ , then $2.00 + 20 = 22.00$ .
$x = 0 \Rightarrow y = 20.00, \quad x = 20 \Rightarrow y = 21.00, \quad x = 40 \Rightarrow y = 22.00, \quad x = 60 \Rightarrow y = 23.00, \quad x = 80 \Rightarrow y = 24.00$
Step 3 — Verify the table is linear
Check that $y$ increases by the same amount for each equal step in $x$ . Here, each increase of 20 in $x$ produces an increase of 1.00 in $y$ . The rate of change per text is $\frac{1.00}{20} = 0.05$ . Constant — confirmed linear.
$\frac{\Delta y}{\Delta x} = \frac{1.00}{20} = 0.05$
Step 4 — Write the equation
Place the slope $m = 0.05$ and the vertical intercept $b = 20$ into the form $y = mx + b$ .
$y = 0.05x + 20$
Step 5 — Interpret slope and vertical intercept in context
The slope $0.05$ means the monthly cost increases by CAD 0.05 for every additional text message. The vertical intercept $20$ means the cost is CAD 20.00 even when zero texts are sent — this is the flat monthly fee.
m = 0.05 (CAD per text), b = 20 (CAD flat fee)
Step 6 — Use the equation to find cost at 150 messages
Substitute $x = 150$ into the equation and evaluate. Multiply first, then add.
$y = 0.05(150) + 20 = 7.50 + 20 = 27.50$

Answer: The equation is

y = 0.05x + 20

. The slope is

0.05

(cost per text) and the vertical intercept is

20

(flat fee). Sending 150 messages costs CAD 27.50.

Check: Use the table pattern to verify

x = 80

gives

y = 24.00

:

0.05 \times 80 + 20 = 4.00 + 20 = 24.00

. ✓ Use two table rows to re-calculate the slope:

\frac{24.00 - 20.00}{80 - 0} = \frac{4.00}{80} = 0.05

. ✓

Common mistakes and how to avoid them

Confusing the slope with the vertical intercept in the equation

y = mx + b

. Students sometimes read the first number as

b

and the constant as

m

.

Correction: Remember:

m

is always the number multiplied by

x

, and

b

is always the stand-alone number added at the end. In

y = 3x + 7

, the slope is

3

and the vertical intercept is

7

.

Calculating the rate of change by dividing the change in

x

by the change in

y

— getting the fraction upside down.

Correction: Rate of change = change in

y

÷ change in

x

. Always put the

y

-change on top:

\frac{\Delta y}{\Delta x}

.

Assuming a table is linear after checking only one pair of consecutive differences.

Correction: Check every consecutive pair of

y

-differences. All gaps between

y

-values must be equal for the same step size in

x

. One matching pair is not enough.

Plotting points as

(y, x)

instead of

(x, y)

on the graph.

Correction: Always write and plot ordered pairs as

(x, y)

: the

x

-value goes first (horizontal axis) and the

y

-value goes second (vertical axis).

Forgetting to include the vertical intercept when writing an equation from a table or graph, writing

y = mx

instead of

y = mx + b

.

Correction: Always check the value of

y

when

x = 0

. If it is not zero, you must include

b

in the equation. Leaving it out shifts the entire line to the wrong position.

Lesson summary

A linear relation has a constant rate of change, which means its graph is always a straight line.
The same linear relation can be shown as a table of values, a graph, or an equation — all three carry identical information.
The slope ( $m$ ) equals the constant rate of change; it appears as equal differences in $y$ in the table, as the steepness of the line in the graph, and as the coefficient of $x$ in the equation.
The vertical intercept ( $b$ ) is the initial value when $x = 0$ ; it appears in the table row where $x = 0$ , as the point where the line crosses the $y$ -axis, and as the constant term in the equation $y = mx + b$ .
To move between representations, identify $m$ and $b$ first — they are the bridge connecting all three forms.
Choose the representation that best fits the question: tables for specific values, graphs for visual trends, equations for calculating any output quickly.

Check your understanding

Question 1

A table of values shows these pairs:

(0, 8)

,

(1, 11)

,

(2, 14)

,

(3, 17)

. Which equation matches this table?

$y = 8x + 3$
$y = 3x + 8$
$y = 3x + 11$
$y = 11x + 3$

Show answer and explanation

y = 3x + 8

The

y

-value when

x = 0

is

8

, so the vertical intercept is

b = 8

. Each time

x

increases by 1,

y

increases by 3, so the slope is

m = 3

. Substituting into

y = mx + b

gives

y = 3x + 8

.

Question 2

A straight-line graph crosses the vertical axis at

y = -4

and rises 5 units for every 1 unit moved to the right. What is the equation of this line?

$y = -4x + 5$
$y = 5x - 4$
$y = -4x - 5$
$y = 4x + 5$

Show answer and explanation

y = 5x - 4

The vertical intercept is

-4

, so

b = -4

. The slope is

\frac{5}{1} = 5

, so

m = 5

. The equation is

y = 5x + (-4)

, written as

y = 5x - 4

.

Question 3

The equation

y = -2x + 10

is given. Which statement correctly describes this relation?

The slope is $10$ and the line rises from left to right.
The slope is $-2$ and the line rises from left to right.
The slope is $-2$ and the line falls from left to right.
The slope is $2$ and the vertical intercept is $-10$ .

Show answer and explanation

The slope is

-2

and the line falls from left to right.

In

y = mx + b

,

m = -2

and

b = 10

. A negative slope means

y

decreases as

x

increases, so the line falls from left to right. The vertical intercept is

10

, not

-10

.

Question 4

You are given the equation

y = 4x - 3

. Which value of

y

belongs in the table when

x = 5

?

$y = 7$
$y = 17$
$y = 23$
$y = 2$

Show answer and explanation

y = 17

Substitute

x = 5

:

y = 4(5) - 3 = 20 - 3 = 17

. The correct value is

y = 17

.

Key terms

Linear relation: A relationship between two variables where the rate of change is constant, producing a straight-line graph.
Table of values: An organised list of input ( $x$ ) and output ( $y$ ) pairs that describes a relation.
Rate of change: How much the output ( $y$ ) changes for each 1-unit increase in the input ( $x$ ). Calculated as change in $y$ divided by change in $x$ .
Initial value: The value of $y$ when $x = 0$ . It is where the relation starts before any change in $x$ occurs.
Slope: The steepness and direction of a line on a graph, equal to rise divided by run, or change in $y$ divided by change in $x$ .
Vertical intercept: The point where a line crosses the vertical ( $y$ ) axis on a graph. It occurs at $x = 0$ and equals the initial value.
Slope-intercept form: The equation form $y = mx + b$ , where $m$ is the slope and $b$ is the vertical intercept.
Ordered pair: A pair of numbers written as $(x, y)$ that gives the location of one point on a graph.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation C3.2. It is a study resource, not an official curriculum publication.

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C3.2 · Connect representations of linear relations

Ontario Grade 9 Mathematics

Moving Between Tables, Graphs, and Equations

What you will learn

What Is a Linear Relation? A Quick Review

The Table of Values Representation

The Graph Representation

The Equation Representation

Choosing the Most Useful Representation

Yard-Work Pay: Table of Values

Three Representations at a Glance

Moving Between All Three Representations

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MTH1W

About this lesson and its review