Graphing Real-World Relationships

Graphs are powerful tools for representing relationships between quantities in the real world. They allow us to see how one variable changes as another variable changes.

Linear graphs are especially useful when a quantity changes at a constant rate.

Examples include:

• money earned at a constant hourly rate
• distance travelled at constant speed
• water entering or leaving a tank at a constant rate
• savings increasing by the same amount each week
• temperature changing steadily over time
• costs that include a fixed fee and a constant charge per unit

A linear model can often be written as:

y = mx + b

where:

• m represents the rate of change
• b represents the starting or initial value

The important skill is not simply drawing the graph. We must also understand what the graph means in the real situation.

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Variables in Real-World Graphs

A graph normally compares two variables.

The independent variable is usually placed on the x-axis.

The dependent variable is usually placed on the y-axis.

For example, suppose we are studying the distance travelled by a car.

We might use:

x-axis = time in hours

y-axis = distance in kilometres

Distance depends on how much time has passed, so time is the independent variable and distance is the dependent variable.

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Labels and Units Matter

A real-world graph should clearly identify what each axis represents.

For example:

x-axis: Time

y-axis: Distance (km)

Without labels, the point:

(3, 180)

has little meaning.

With the labels, we can interpret it:

After 3 hours, the car has travelled 180 km.

Always include units when interpreting real-world graphs.

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Representing Real-World Data

Suppose a cyclist travels at a constant speed of:

20 km/h

The data might be:

These values can be plotted as coordinate pairs:

(0, 0)

(1, 20)

(2, 40)

(3, 60)

(4, 80)

(5, 100)

Because the cyclist travels at a constant rate, the points form a straight-line pattern.

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Finding the Rate of Change

The rate of change can be calculated using:

m = Δy/Δx

Using:

(1, 20)

and:

(4, 80)

we get:

m = (80 − 20)/(4 − 1)

m = 60/3

m = 20

In context:

m = 20 km/h

The cyclist travels an additional 20 km for every hour.

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Writing the Equation

The graph passes through:

(0, 0)

so:

b = 0

The slope is:

m = 20

Using:

y = mx + b

we get:

d = 20t

where:

d = distance in kilometres

t = time in hours

The equation, table, and graph all represent the same real-world relationship.

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Interpreting the Slope

In a real-world graph, slope should be interpreted using the quantities and units on the axes.

For a distance-time graph:

slope = kilometres/hour

For a cost-items graph:

slope = dollars/item

For a volume-time graph:

slope = litres/minute

For a temperature-time graph:

slope = °C/hour

A complete interpretation explains both the numerical value and its meaning.

Instead of:

"The slope is 15."

write:

"The distance increases by 15 km for every hour that passes."

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Interpreting the y-Intercept

The y-intercept represents the value of y when:

x = 0

In real-world situations, it often represents the starting value.

For example:

C = 4d + 8

where:

C = taxi cost

d = distance travelled

The y-intercept is:

8

This means:

The taxi costs $8 before any distance is travelled.

It could represent a starting or booking fee.

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Real-World Example: Taxi Fare

Suppose a taxi charges:

• $6 starting fee
• $2.50 per kilometre

The equation is:

C = 2.5d + 6

Create a table:

The graph would be a straight line with:

slope = 2.5 dollars/km

and:

y-intercept = $6

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Interpreting Points on the Taxi Graph

Consider:

(6, 21)

The x-coordinate represents distance.

The y-coordinate represents cost.

Therefore:

A 6 km taxi journey costs $21.

A good interpretation names both variables and includes the appropriate units.

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Identifying Trends

A trend describes the general pattern shown by data.

A graph may show:

• an increasing trend
• a decreasing trend
• no change
• a changing or non-linear trend

For a linear graph, the trend has a constant rate of change.

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Increasing Trends

A graph that rises from left to right has a positive slope.

This means that as x increases:

y increases

Examples include:

• savings increasing over time
• distance increasing with time
• earnings increasing with hours worked
• total cost increasing with items purchased

The steeper the line, the greater the rate of increase, assuming the graph scales are the same.

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Decreasing Trends

A graph that falls from left to right has a negative slope.

This means that as x increases:

y decreases

Examples include:

• water draining from a tank
• fuel being consumed
• a candle becoming shorter
• temperature decreasing during cooling
• battery charge decreasing with use

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No Change

A horizontal graph has:

slope = 0

This means the dependent variable remains constant.

For example, suppose the water level in a container remains at:

50 cm

for several minutes.

The graph would be horizontal.

The rate of change is:

0 cm/min

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Creating an Equation from a Graph

Suppose a graph contains:

(0, 10)

and:

(4, 30)

First calculate slope:

m = (30 − 10)/(4 − 0)

m = 20/4

m = 5

The y-intercept is:

10

Therefore:

y = 5x + 10

If the graph represents money earned over time, the equation might be:

E = 5t + 10

The context determines the variable names and units.

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Creating an Equation When the Intercept Is Not Obvious

Suppose a line contains:

(2, 11)

and:

(6, 23)

Calculate slope:

m = (23 − 11)/(6 − 2)

m = 12/4

m = 3

Use:

y = 3x + b

Substitute:

(2, 11)

11 = 3(2) + b

11 = 6 + b

b = 5

Therefore:

y = 3x + 5

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Real-World Example: Savings

Suppose Mia begins with:

$120

and saves:

$30 each week

The equation is:

S = 30w + 120

where:

S = savings

w = weeks

The graph has:

slope = $30/week

and:

y-intercept = $120

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Making Predictions

Once we have a linear model, we can use it to predict values.

Using:

S = 30w + 120

predict Mia's savings after 12 weeks.

S = 30(12) + 120

S = 360 + 120

S = $480

The model predicts that Mia will have $480 after 12 weeks.

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Interpolation

Interpolation means estimating a value within the range of known data.

Suppose data has been collected between:

0 and 10 hours.

Estimating the value at:

6.5 hours

would be interpolation.

Interpolation is often reasonably reliable when the data follows a strong linear pattern.

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Extrapolation

Extrapolation means predicting a value outside the range of known data.

Suppose measurements were collected between:

0 and 10 hours.

Predicting the value at:

25 hours

would be extrapolation.

Extrapolation is generally less reliable because the original trend may not continue indefinitely.

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Why Extrapolation Can Be Risky

Suppose a plant grows approximately:

2 cm/week

during its first six weeks.

It would be unreasonable to assume that it will continue growing at exactly 2 cm/week forever.

Environmental conditions may change.

Growth may slow.

The plant may reach maturity.

Therefore, a linear model that works well over one interval may become inaccurate when extended too far.

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Real-World Example: Draining Tank

A tank initially contains:

500 L

and drains at:

25 L/min

Equation:

V = 500 − 25t

where:

V = volume in litres

t = time in minutes

The slope is:

−25 L/min

The intercept is:

500 L

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Predicting When the Tank Is Empty

Set:

V = 0

Then:

0 = 500 − 25t

25t = 500

t = 20

The model predicts that the tank becomes empty after:

20 minutes

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Recognizing the Model's Limit

If we substitute:

t = 30

we get:

V = 500 − 25(30)

V = 500 − 750

V = −250 L

Mathematically, the equation produces:

−250 L

But a tank cannot contain negative water.

Therefore, the model is only realistic until:

t = 20 minutes

A mathematically correct calculation can still produce an unreasonable real-world conclusion.

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Domain Restrictions

A domain restriction limits the x-values that make sense in a real situation.

For the draining tank:

0 ≤ t ≤ 20

This means the model applies from the start until the tank becomes empty.

Real-world graphs often have more restricted domains than their mathematical equations.

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Discrete and Continuous Data

Some real-world relationships are continuous.

Examples:

• time
• distance
• temperature
• mass
• volume

These quantities can take values between whole numbers.

Other situations are discrete.

Examples:

• number of students
• number of cars
• number of books
• number of tickets

You might buy:

3 tickets

or:

4 tickets

but normally not:

3.6 tickets

This affects how the graph should be interpreted.

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Real-World Example: Ticket Sales

Suppose tickets cost:

$12 each

Revenue is:

R = 12n

where:

n = number of tickets

Possible points include:

(0, 0)

(1, 12)

(2, 24)

(3, 36)

Because tickets are counted as whole objects, the meaningful values of n are whole numbers.

Although the equation forms a mathematical line, the realistic situation consists of discrete points.

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Comparing Two Real-World Models

Suppose two phone plans have costs:

Plan A: C = 4d + 20

Plan B: C = 2d + 40

Plan A has:

slope = 4

intercept = 20

Plan B has:

slope = 2

intercept = 40

Plan A starts cheaper but increases more quickly.

Plan B starts more expensive but increases more slowly.

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Finding the Break-Even Point

Set the costs equal:

4d + 20 = 2d + 40

Subtract 2d:

2d + 20 = 40

Subtract 20:

2d = 20

d = 10

Find the cost:

C = 4(10) + 20

C = 60

The plans have the same cost at:

(10, 60)

This point is called the intersection or break-even point.

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Interpreting an Intersection

The intersection of two graphs represents a point where both relationships have the same:

• x-value
• y-value

For the phone plans:

(10, 60)

means:

At 10 units of usage, both plans cost $60.

Before and after the intersection, one plan may be more economical than the other.

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Comparing Trends

Suppose two savings graphs have slopes:

Plan A:

25 dollars/week

Plan B:

40 dollars/week

Plan B has the greater rate of increase.

However, this does not necessarily mean Plan B always has more money.

Plan A might have started with a much larger amount.

Both slope and intercept should be considered.

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Graphs and Predictions

Graphs can be used to make predictions by:

• reading values directly
• extending a linear trend
• using an equation developed from the graph

Equations usually provide more precise predictions than visually estimating from a graph.

Graphs are especially useful for understanding the overall pattern.

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Exact Values and Estimated Values

Suppose a graph suggests that:

x = 7

corresponds to approximately:

y = 42

If the graph is drawn from an exact equation, the equation may provide a more precise value.

Graphs may be limited by:

• line thickness
• plotting accuracy
• graph scale
• reading error

Therefore, graphical values are often described as estimates unless exact coordinates are clearly known.

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Data Does Not Always Form a Perfect Line

Real-world measurements often contain variation.

For example:

The values do not increase by exactly the same amount every hour.

However, they show a general increasing trend.

A linear model may still be useful if the data points are reasonably close to a straight-line pattern.

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Linear Models and Real Data

A linear model is a straight-line equation used to approximate a relationship.

The data itself may not be perfectly linear.

A model attempts to capture the general pattern.

For example:

T = 2.2h + 10

might approximate temperature data.

The equation can then be used to:

• describe the trend
• estimate values
• make predictions
• compare rates

But the predictions should not automatically be treated as exact measurements.

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Line of Best Fit

When real data points do not lie exactly on one line, a line of best fit may be used to represent the general trend.

A good line of best fit passes through the middle of the data pattern rather than attempting to connect every point.

The line can then be used to estimate the relationship between the variables.

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Correlation and Causation

If two variables show a relationship on a graph, this does not automatically mean that one variable causes the other.

For example, both:

• ice cream sales
• swimming activity

may increase during hot weather.

A graph could show a relationship between them.

But buying ice cream does not necessarily cause people to swim.

A third factor—temperature—may influence both.

This is why graphical conclusions must be evaluated carefully.

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Positive Correlation

A data set shows positive correlation when larger x-values tend to be associated with larger y-values.

The data generally rises from left to right.

Examples might include:

• study time and test performance
• height and arm span
• advertising and sales under some conditions

Positive correlation does not necessarily prove causation.

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Negative Correlation

A data set shows negative correlation when larger x-values tend to be associated with smaller y-values.

Examples might include:

• speed and travel time for a fixed distance
• age of some products and resale value
• time and remaining fuel during steady consumption

Again, the relationship must be interpreted in context.

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No Clear Correlation

Sometimes the points show no obvious upward or downward pattern.

In this case, there may be:

little or no correlation

A linear model may not be useful.

Forcing a straight-line model onto data with no meaningful linear trend can produce misleading conclusions.

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Evaluating Whether a Linear Model Is Appropriate

Before using a linear model, ask:

• Does the data show an approximately straight-line pattern?
• Is the rate reasonably constant?
• Is the prediction within or far beyond the measured data?
• Does the predicted value make physical sense?
• Are there natural limits?
• Are the units appropriate?
• Is the data continuous or discrete?
• Are there unusual points affecting the trend?

A model should be mathematically useful and reasonable in context.

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Outliers

An outlier is a data point that lies unusually far from the general pattern.

Suppose most values follow a steady increasing trend but one point is much higher than expected.

Possible reasons include:

• measurement error
• recording error
• unusual conditions
• genuine variation

An outlier should not automatically be deleted.

Instead, investigate why it differs from the rest of the data.

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Graph Scale and Misleading Conclusions

The appearance of a graph can change dramatically depending on its scale.

A small change can appear enormous if the y-axis covers a very narrow range.

The same change may appear tiny if the y-axis covers a very large range.

When evaluating a graph, examine:

• axis labels
• units
• minimum and maximum values
• interval sizes
• whether an axis begins at zero
• whether part of an axis has been omitted

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A Truncated Axis

A graph does not always need to begin at zero.

Sometimes starting near the data values helps reveal small changes.

However, this can also make differences appear larger than they really are.

For example, values:

98, 99, 100, 101

may look dramatically different on a graph ranging from:

97 to 102

but much less dramatic on a graph ranging from:

0 to 110.

Always examine the scale before drawing conclusions.

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Evaluating a Prediction

Suppose a business has increased sales by approximately:

$5,000 per month

for six months.

A linear model predicts continued growth for the next 20 years.

Is that conclusion reasonable?

Probably not without additional evidence.

Markets change.

Businesses encounter limits.

Prices change.

Competitors enter the market.

A model that works over six months may not remain valid for decades.

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Mathematical Accuracy vs Real-World Reasonableness

A prediction can be mathematically correct according to a model but unrealistic in context.

For example:

A child grows 6 cm in one year.

A model that assumes:

6 cm/year forever

would predict an enormous adult height after many decades.

The arithmetic is correct.

The model assumption is not.

This distinction is essential when using mathematics to describe the real world.

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Interpreting Negative Values

Suppose a model predicts:

−20 customers

A negative customer count is impossible.

This suggests that:

• the model has been extended too far
• the domain should be restricted
• a different model may be required

Always ask whether the output is physically or logically possible.

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Evaluating Discrete Predictions

Suppose a model predicts that a school needs:

4.6 buses

A school cannot normally use 0.6 of a bus.

If all students must be transported, the result would need to be interpreted as:

5 buses

Real-world interpretation sometimes requires more than simply reporting the calculated decimal.

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Reliability of Interpolation and Extrapolation

In general:

Interpolation is usually more reliable than extrapolation.

Why?

Interpolation stays within the region where data has already been observed.

Extrapolation assumes the pattern continues beyond the observed data.

The farther we extrapolate, the greater the risk that conditions will change.

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A Complete Worked Example

A water tank initially contains:

100 L

Water enters at a constant rate of:

12 L/min

Equation:

V = 12t + 100

where:

V = volume in litres

t = time in minutes

Create a table:

The slope is:

12 L/min

The y-intercept is:

100 L

The point:

(6, 172)

means:

After 6 minutes, the tank contains 172 L.

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Evaluating the Tank Model

Suppose the tank can hold a maximum of:

250 L

Find when the model reaches this capacity.

250 = 12t + 100

150 = 12t

t = 12.5

The tank reaches capacity after:

12.5 minutes

The equation could mathematically predict larger volumes after this time.

However, if the tank overflows, the model no longer accurately describes the amount of water inside the tank.

The realistic domain of the model must therefore consider the tank's capacity.

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From Graph to Equation to Prediction

A powerful modelling process is:

graph → slope and intercept → equation → prediction → evaluation

For example:

1. examine a graph
2. calculate the rate of change
3. identify the starting value
4. write y = mx + b
5. use the equation to predict a value
6. decide whether that prediction makes sense

The final step is essential.

Mathematical modelling is not complete until the result is interpreted.

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A Reliable Strategy for Analyzing Real-World Graphs

When analyzing a real-world linear relationship:

1. Identify the variables.
2. Check the axis labels.
3. Check the units.
4. Check the scales.
5. Identify the overall trend.
6. Determine whether the relationship appears linear.
7. Choose two useful points.
8. Calculate the slope.
9. Interpret the slope with units.
10. Identify the y-intercept.
11. Interpret the intercept in context.
12. Write a linear equation if appropriate.
13. Use the model to make predictions.
14. Decide whether the prediction is interpolation or extrapolation.
15. Check whether the result is realistic.
16. Identify any domain restrictions or limitations.

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Common Mistakes

Ignoring units

A slope of 20 is incomplete in a real-world problem if it should be:

20 km/h

or:

$20/week

Describing the slope without context

"The slope is −5" is less informative than:

"The temperature decreases by 5°C each hour."

Confusing slope and intercept

Slope describes change.

Intercept describes the value when x = 0.

Assuming every trend is linear

Some relationships curve, level off, accelerate, or fluctuate.

Connecting every data point

Real measured data may vary. A line of best fit is often more useful than connecting each point.

Assuming correlation proves causation

Variables can be related without one directly causing the other.

Extrapolating too far

A trend observed over a short period may not continue indefinitely.

Ignoring physical limits

Negative populations, negative volumes, or values above maximum capacity may be impossible.

Ignoring graph scale

A graph can visually exaggerate or minimize differences.

Treating predictions as exact measurements

A model provides estimates based on assumptions.

Ignoring discrete quantities

Predictions involving people, buses, tickets, or other countable objects may need appropriate whole-number interpretation.

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Key Terms

Real-world relationship — mathematical connection between quantities observed or defined in a practical situation.

Linear relationship — relationship with a constant rate of change.

Linear model — straight-line equation used to represent or approximate a relationship.

Independent variable — input variable, usually placed on the x-axis.

Dependent variable — output variable, usually placed on the y-axis.

Slope — rate of change of the dependent variable relative to the independent variable.

y-intercept — value of the dependent variable when the independent variable equals zero.

Trend — general direction or pattern shown by data.

Prediction — estimate of an unknown or future value based on a model or pattern.

Interpolation — estimating a value within the range of observed data.

Extrapolation — predicting a value outside the range of observed data.

Domain — set of possible or meaningful input values.

Domain restriction — limitation placed on input values because of mathematical or real-world conditions.

Continuous data — data that can take any value within an interval.

Discrete data — data consisting of separate countable values.

Line of best fit — line used to represent the overall linear trend in a set of data.

Correlation — relationship or association between two variables.

Positive correlation — general tendency for y to increase as x increases.

Negative correlation — general tendency for y to decrease as x increases.

Outlier — data point unusually far from the overall pattern.

Break-even point — point where two quantities, often costs or revenues, are equal.

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Key Takeaways

• Real-world relationships can often be represented using graphs.
• Linear graphs model situations with approximately constant rates of change.
• The x-axis normally represents the independent variable.
• The y-axis normally represents the dependent variable.
• Axis labels, scales, and units are essential for interpreting real-world graphs.
• Slope represents the rate at which one quantity changes relative to another.
• The units of slope are y-units per x-unit.
• Positive slope represents an increasing trend.
• Negative slope represents a decreasing trend.
• Zero slope represents no change.
• The y-intercept often represents an initial or starting value.
• A linear equation can be created from a graph by determining its slope and intercept.
• Linear models can be used to make predictions.
• Interpolation predicts within the observed data range.
• Extrapolation predicts beyond the observed data range.
• Extrapolation is generally less reliable than interpolation.
• Real data does not always fall exactly on a straight line.
• A line of best fit can approximate an overall linear trend.
• Correlation does not automatically mean causation.
• Graph scale can influence how dramatic a trend appears.
• Predictions should always be evaluated for real-world reasonableness.
• Mathematical models have limitations and should only be used within appropriate conditions.
• A mathematically correct answer may still be unrealistic in context.
• Good mathematical modelling includes representing, calculating, predicting, interpreting, and evaluating.

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Check Your Understanding

1. A graph shows time in hours on the x-axis and distance in kilometres on the y-axis. Explain what a slope of 65 means in context.
2. A taxi fare graph has a y-intercept of $7. What does this most likely represent?
3. A savings account begins with $200 and increases by $25 each week. Write a linear equation and explain the meaning of its slope and intercept.
4. A line passes through (0, 8) and (5, 28). Calculate the slope and write its equation in slope-intercept form.
5. A tank initially contains 400 L and loses 20 L each minute. Create an equation modelling the volume of water remaining after t minutes.
6. Using your model from Question 5, determine when the tank becomes empty. Explain why the model should not be used to predict the tank's volume after this time.
7. A cyclist's distance is represented by d = 18t + 5. Interpret both 18 and 5 in a possible real-world context.
8. A model was created using data collected between x = 0 and x = 20. Explain the difference between predicting y when x = 12 and predicting y when x = 50. Which prediction would generally be more reliable?
9. Two phone plans are represented by A = 3x + 15 and B = 2x + 30. Determine where the two plans have the same cost and interpret the intersection in context.
10. A student says, "This graph rises from left to right, so the two variables must have a cause-and-effect relationship." Explain why this conclusion is not necessarily correct.
11. A company records sales for six months and observes an approximately linear increase of $4,000 per month. Explain why using the same model to predict sales 30 years later could be unreliable.
12. A linear model predicts that a school will require 3.4 buses for a trip. Explain how this result should be interpreted if every student must have transportation.
13. A graph of temperature against time has a slope of −2.5°C/min and a y-intercept of 80°C. Write an equation for the relationship and explain both values in context.
14. A graph shows values increasing from 98 to 102, but the y-axis begins at 97 instead of 0. Explain how this choice of scale could affect a viewer's impression of the change.
15. A water tank initially contains 150 L, fills at 10 L/min, and has a maximum capacity of 500 L. Write a linear model, determine when the tank reaches capacity, explain the meaningful domain of the model, and discuss why continuing the linear graph beyond that point could lead to an unreasonable conclusion.