The graph visually shows that:

• the first relationship starts higher
• the second has a steeper slope
• the two relationships eventually meet and then change order

This demonstrates why graphs are useful for comparisons.

Comparing a Table with an Equation

Suppose Plan A is:

A = 4x + 12

Plan B is represented by:

From the table:

slope = 6

intercept = 5

Therefore:

B = 6x + 5

Now comparison is easier.

Plan A:

m = 4, b = 12

Plan B:

m = 6, b = 5

Plan A starts higher.

Plan B increases faster.

Comparing an Equation with Words

Suppose:

Plan A: C = 8h + 20

Plan B is described as:

"A $10 starting fee plus $12 per hour."

Translate Plan B:

C = 12h + 10

Now compare:

Plan A:

rate = $8/hour

starting fee = $20

Plan B:

rate = $12/hour

starting fee = $10

The equation form makes the mathematical comparison easier.

Comparing Two Tables

Relationship A

Relationship B

Relationship A:

m = 3

b = 10

Relationship B:

m = 5

b = 2

Therefore:

• A has the larger starting value
• B has the larger rate of change

Using a Common Representation

When comparing relationships shown in different forms, it is often helpful to convert them to a common representation.

For example:

• convert both to equations
• convert both to tables
• graph both relationships together

Equations are especially efficient for comparing slope and intercept.

Graphs are especially effective for comparing overall behaviour and intersections.

Comparing Direct Variation Representations

Recall that direct variation has the form:

y = kx

Suppose a table shows:

The constant ratio is:

y/x = 4

Therefore:

y = 4x

A verbal description might be:

"The output is four times the input."

A graph would be a straight line through:

(0, 0)

with slope 4.

All representations describe the same proportional relationship.

Distinguishing Proportional and Non-Proportional Linear Relationships

Compare:

A: y = 4x

and:

B: y = 4x + 6

Both are linear.

Both have slope:

4

But only A is proportional.

Why?

A has:

b = 0

and passes through:

(0, 0)

B has:

b = 6

and does not pass through the origin.

Representations can help reveal this distinction.

Real-World Example: Two Taxi Companies

Taxi A charges:

$5 starting fee + $3/km

Therefore:

A = 3d + 5

Taxi B charges:

$9 starting fee + $2/km

Therefore:

B = 2d + 9

Taxi A:

slope = 3

intercept = 5

Taxi B:

slope = 2

intercept = 9

Taxi A starts cheaper but increases more quickly.

Taxi B starts more expensive but increases more slowly.

Finding the Break-Even Distance

Set the taxi costs equal:

3d + 5 = 2d + 9

Subtract 2d:

d + 5 = 9

d = 4

At 4 km:

A = 3(4) + 5 = 17

B = 2(4) + 9 = 17

Therefore, both taxis cost:

$17 at 4 km

For shorter journeys, Taxi A is cheaper.

For longer journeys, Taxi B is cheaper.

Real-World Example: Savings Plans

Plan A begins with:

$200

and adds:

$25/week

Equation:

A = 25w + 200

Plan B begins with:

$80

and adds:

$40/week

Equation:

B = 40w + 80

Plan A has the greater starting value.

Plan B has the greater rate of change.

Different representations can help answer different questions about these plans.

Which Representation Is Best?

There is no single representation that is always best.

The most useful representation depends on what you want to know.

Strong mathematical reasoning involves choosing the representation that best suits the problem.

When a Table Is Most Useful

Suppose a scientist records temperature every hour.

A table is useful because it preserves the exact measured values.

It allows the scientist to:

• organize observations
• compare specific measurements
• identify possible patterns

However, a table may not make the overall trend as obvious as a graph.

When a Graph Is Most Useful

Suppose you want to compare two savings plans over 20 weeks.

A graph quickly reveals:

• which starts higher
• which grows faster
• when one overtakes the other

This information may be less immediately obvious from a long table.

 

 

 

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When an Equation Is Most Useful

Suppose you need the exact cost of a 37 km journey.

If:

C = 2.5d + 6

then:

C = 2.5(37) + 6

C = 92.5 + 6

C = $98.50

An equation allows exact calculation without needing a table containing d = 37.

When Words Are Most Useful

Consider:

y = −5x + 100

Without context, we know its mathematical structure but not what it represents.

Now suppose we are told:

"A tank initially contains 100 L and drains at 5 L per minute."

The verbal description gives meaning to:

−5 → water decreases by 5 L/min

100 → initial volume

Words provide the context that the equation alone cannot.

Strengths and Limitations of Tables

Strengths

• organized
• easy to read
• shows exact listed values
• useful for recognizing patterns
• useful for recorded data

Limitations

• shows only selected values
• may become very long
• trends may be difficult to see
• values between rows may not be obvious

Strengths and Limitations of Graphs

Strengths

• shows overall behaviour
• makes trends visible
• makes comparisons easier
• shows intersections visually
• shows increasing and decreasing behaviour

Limitations

• exact values may be difficult to read
• accuracy depends on scale
• poor graph scales can be misleading

Strengths and Limitations of Equations

Strengths

• compact
• precise
• allows exact calculations
• useful for predictions
• clearly shows mathematical structure

Limitations

• may not immediately show overall behaviour to a beginner
• may not explain the real-world meaning
• requires algebraic understanding

Strengths and Limitations of Verbal Descriptions

Strengths

• provides context
• explains units
• explains meaning
• connects mathematics to real situations

Limitations

• can be longer than an equation
• mathematical structure may be less obvious
• wording can sometimes be ambiguous

Checking Whether Representations Agree

Suppose an equation is:

y = 6x − 2

and a table claims to represent it:

Check:

x = 0:

6(0) − 2 = −2

x = 1:

6(1) − 2 = 4

x = 2:

6(2) − 2 = 10

The values agree.

Therefore, the representations are equivalent.

Finding an Error Between Representations

Suppose:

y = 5x + 3

and a table shows:

Check x = 3:

5(3) + 3 = 18

But the table says:

19

Therefore, the final row does not match the equation.

Comparing representations is a useful way to detect errors.

Analyzing Rate of Change Across Representations

Suppose three relationships are given.

A

y = 4x + 2

B

"The value begins at 10 and increases by 6 for every unit of x."

C

Rates:

A:

4

B:

6

C:

m = (13 − 3)/(2 − 0)

m = 10/2

5

Therefore, from greatest to least rate:

B, C, A

Analyzing Starting Values Across Representations

Using the same relationships:

A:

b = 2

B:

starting value = 10

C:

when x = 0:

y = 3

Therefore, from greatest to least starting value:

B, C, A

The representation changes, but the mathematical features can still be compared.

Representations Can Reveal Different Features

Consider:

y = −2x + 12

The equation immediately reveals:

slope = −2

intercept = 12

A table makes specific values easy to read.

A graph makes the decreasing trend obvious.

A verbal description can explain what −2 and 12 mean in context.

The representations are not competing versions of the mathematics. They are different ways of viewing the same relationship.

A Reliable Strategy for Comparing Representations

When comparing tables, graphs, equations, and descriptions:

1. Identify the variables.
2. Determine whether the relationship is linear.
3. Find the rate of change.
4. Find the starting value or y-intercept.
5. Record the units if the situation has context.
6. Convert representations if necessary.
7. Compare slopes.
8. Compare intercepts.
9. Check whether the representations produce the same input-output pairs.
10. Decide whether they are equivalent.
11. Choose the representation best suited to the question.
12. Explain what the mathematical features mean in context.

Common Mistakes

Assuming different-looking representations describe different relationships

A table, equation, graph, and description may all represent exactly the same relationship.

Comparing only one point

Two different lines can share one point. Equivalent linear relationships must have the same slope and intercept.

Comparing only the intercept

Two relationships can start at the same value but change at different rates.

Comparing only the slope

Two parallel relationships can have the same slope but different intercepts.

Using Δy without considering Δx

Rate of change is:

Δy/Δx

Confusing a larger intercept with a larger rate

The intercept describes the starting value. The slope describes how quickly the relationship changes.

Assuming the graph is always the most accurate representation

Graphs are excellent for visualization, but equations and tables may provide more precise numerical values.

Ignoring units

A rate of 5 could mean $5/hour, 5 km/s, or 5 L/min.

Assuming every straight line represents direct variation

A direct variation must also pass through the origin.

Failing to check converted representations

After converting a table or description to an equation, substitute values to verify the result.

Key Terms

Representation — way of showing or communicating a mathematical relationship.

Table — organized display of input-output values.

Graph — visual representation of the relationship between variables.

Equation — mathematical statement describing a relationship between quantities.

Verbal description — explanation of a mathematical relationship using words.

Equivalent representations — different representations describing the same mathematical relationship.

Slope — rate of change of a linear relationship.

Rate of change — change in one variable compared with the change in another.

y-intercept — value of y when x = 0.

Starting value — initial value of the dependent variable.

Linear relationship — relationship with a constant rate of change.

Direct variation — proportional linear relationship of the form y = kx.

Input — independent variable value.

Output — dependent variable value.

Intersection — point where two graphs meet and therefore have the same input and output values.

Conversion — process of changing one representation into another while preserving the same relationship.

Key Takeaways

• Mathematical relationships can be represented using tables, graphs, equations, and words.
• Different representations can describe exactly the same relationship.
• Equivalent linear representations have the same slope and y-intercept.
• Tables are useful for displaying exact selected values and identifying patterns.
• Graphs are useful for visualizing trends, rates, and intersections.
• Equations are useful for exact calculations, predictions, and algebraic analysis.
• Verbal descriptions provide context, meaning, and units.
• Linear equations can be written in the form y = mx + b.
• The slope m appears as a constant rate in tables, steepness on graphs, a coefficient in equations, and a rate in verbal descriptions.
• The y-intercept b appears as the value when x = 0, the point where a graph crosses the y-axis, the constant term in an equation, and the starting value in words.
• Converting relationships to a common representation can make comparisons easier.
• A relationship with a greater starting value does not necessarily have a greater rate of change.
• A relationship with a greater rate of change may eventually overtake one that starts higher.
• Intersections represent values for which two relationships have the same output.
• Comparing representations can help detect errors.
• A graph may make behaviour easier to see, while an equation may provide greater numerical precision.
• The best representation depends on the question being investigated.
• Strong mathematical reasoning involves being able to move flexibly between different representations.

Check Your Understanding

1. Describe one advantage of representing a relationship using a table, a graph, an equation, and a verbal description.
2. Write a table containing x = 0, 1, 2, 3, and 4 for:

y = 3x + 5

3. Write a verbal description of:

y = 4x + 7

that clearly identifies the starting value and rate of change.

4. Write an equation for the table:

5. Determine whether the equation y = 5x + 2 is equivalent to the table:

Explain how you know.

6. A relationship is described as: "The value begins at 12 and decreases by 3 for every increase of 1 in x." Write its equation and create a table for x = 0 through x = 4.
7. Compare y = 2x + 15 and y = 6x + 3. Which has the greater starting value? Which has the greater rate of change?
8. Relationship A is y = 4x + 8. Relationship B is represented by:

Write an equation for Relationship B and compare the slopes and intercepts of the two relationships.

9. A taxi company charges a $7 starting fee plus $3 per kilometre. Represent this relationship using words, an equation, and a table for journeys of 0, 1, 2, 3, and 4 km.
10. A savings account begins with $150 and increases by $25 each week. Write an equation, construct a table for the first five weeks, and explain how the slope and intercept appear in each representation.
11. Relationship A is represented by y = 3x + 10. Relationship B begins at 4 and increases by 5 for every unit of x. Determine where the two relationships have the same value.
12. A student claims that two linear relationships are equivalent because both have a slope of 6. Explain why this information alone is not enough. What else must be compared?
13. A table shows the points (0, 4), (2, 10), (4, 16), and (6, 22). Determine the equation, describe the relationship in words, and explain what its graph would look like.
14. Two phone plans are represented differently. Plan A is C = 5g + 20. Plan B has a $35 starting charge and costs an additional $3 per GB. Write both plans as equations, compare their starting costs and rates, and determine the amount of data for which they cost the same.
15. A tank initially contains 500 L and loses 20 L per minute. Represent the situation with an equation, a verbal description, and a table for the first five minutes. Explain what its graph would show, identify the slope and intercept in each representation, and discuss which representation would be most useful for determining exactly when the tank reaches 100 L.