Applications of Linear Functions

Linear functions are useful because many real-world situations involve a quantity that changes at an approximately constant rate.

Examples include:

• wages earned per hour
• distance travelled at constant speed
• taxi fares
• phone plans
• savings
• filling or draining containers
• temperature changes over short periods
• rental costs
• production costs

A common form is:

y = mx + b

where:

• m = rate of change
• b = initial value
• x = independent variable
• y = dependent variable

In applications, the important question is not only What are m and b? but also:

What do they mean in this situation?

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Linear Functions as Models

A mathematical model uses mathematics to represent a real situation.

Suppose a bicycle rental company charges:

• $8 initial fee
• $5 per hour

Let:

C = total cost

h = number of hours

The linear model is:

C = 5h + 8

Here:

m = 5

means the cost increases by $5 per hour.

b = 8

means the customer pays $8 before any hourly charges are added.

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Recognizing a Linear Situation

A relationship is linear when it has a constant rate of change.

Suppose a worker earns $18 per hour.

Every additional hour increases earnings by:

$18

Therefore the rate of change is constant.

The model is:

E = 18h

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Initial Values

Many linear models include an initial value.

Suppose a student already has $100 and saves $25 each week.

Then:

S = 25w + 100

where:

• S = total savings
• w = weeks

The initial value is:

$100

The rate of change is:

$25/week

After 8 weeks:

S = 25(8) + 100

S = 300

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Interpreting Slope in Context

Slope represents:

change in output / change in input

Its meaning depends on the variables.

Examples:

The units of slope are essential to its interpretation.

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Interpreting the y-Intercept

The y-intercept represents the value of y when:

x = 0

Depending on the situation, it might represent:

• starting money
• initial temperature
• starting distance
• fixed charge
• base fee
• initial population
• starting volume

For:

C = 4x + 15

if C represents cost, then:

$15

could represent a fixed initial charge.

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Solving Problems with Linear Functions

A useful strategy is:

Define → Model → Calculate → Interpret → Check

Define

Identify what the variables represent.

Model

Write or identify the linear equation.

Calculate

Substitute the required value or solve for the unknown.

Interpret

Explain the answer in the context of the problem.

Check

Determine whether the answer is mathematically and practically reasonable.

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Predicting Values

Suppose:

C = 6x + 20

models the cost of renting equipment, where x is the number of hours.

Predict the cost for 7 hours.

Substitute:

x = 7

C = 6(7) + 20

C = 42 + 20

C = 62

Therefore:

A 7-hour rental costs $62.

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Working Backwards

Linear models can also be used to determine an input.

Suppose:

C = 6x + 20

and the total cost is $80.

Set:

C = 80

80 = 6x + 20

Subtract 20:

60 = 6x

Therefore:

x = 10

A total cost of $80 corresponds to:

10 hours

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Distance and Constant Speed

Suppose a car begins 30 km from a city and travels away from it at 70 km/h.

Let:

d = distance from the city

t = time in hours

Model:

d = 70t + 30

After 3 hours:

d = 70(3) + 30

d = 240

Therefore:

The car is 240 km from the city after 3 hours.

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Interpreting a Distance-Time Graph

On a distance-time graph:

horizontal axis = time

vertical axis = distance

The slope represents:

speed

A steeper line represents a greater speed.

A horizontal line means:

distance is not changing

so the object is stationary relative to the reference point.

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Positive and Negative Slopes in Context

Suppose:

V = −15t + 300

models water in a tank.

The slope:

−15 L/min

means the tank loses 15 L every minute.

The intercept:

300 L

means the tank initially contains 300 L.

A negative slope does not mean the model is wrong. It represents a decreasing quantity.

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When Will the Tank Be Empty?

Use:

V = −15t + 300

When the tank is empty:

V = 0

Therefore:

0 = −15t + 300

15t = 300

t = 20

The tank becomes empty after:

20 minutes

After that, the model is no longer physically appropriate because it would predict negative water volume.

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Domain in Real-World Models

Mathematically, a linear function may continue indefinitely.

A real-world model often cannot.

For the tank:

V = −15t + 300

a reasonable domain is:

0 ≤ t ≤ 20

because:

• negative time is not relevant
• after 20 minutes the tank is empty

The context determines the meaningful domain.

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Interpreting Graphs

A real-world graph can communicate:

• initial value
• rate of change
• increasing or decreasing behaviour
• points where important events occur
• intersections between models
• ranges where the model makes sense

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Reading an Initial Value from a Graph

Suppose a graph of savings against time crosses the vertical axis at:

$200

This means:

At time 0, the person already had $200.

It does not mean the person saves $200 each week.

The rate of saving comes from the slope, not the intercept.

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Reading Rate from a Graph

Suppose a graph passes through:

(2, 70)

and:

(6, 150)

Slope:

m = (150 − 70)/(6 − 2)

m = 80/4

m = 20

If x represents weeks and y represents dollars:

The quantity increases by $20 per week.

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Constructing a Model from Data

Suppose:

Find the rate:

m = (130 − 90)/(4 − 2)

m = 40/2

m = 20

Use:

S = 20w + b

Substitute:

90 = 20(2) + b

90 = 40 + b

b = 50

Therefore:

S = 20w + 50

The model tells us the person started with $50 and saves $20 per week.

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Prediction Using a Model

Using:

S = 20w + 50

predict the savings after 15 weeks.

S = 20(15) + 50

S = 350

The model predicts:

$350 after 15 weeks.

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Interpolation

Suppose data were collected between:

week 2 and week 10

Using the model to estimate the value at:

week 6

is interpolation.

Interpolation means making a prediction within the range of observed data.

It is often more reliable than predicting far beyond the available data.

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Extrapolation

Using the same data to predict:

week 100

is extrapolation.

This prediction lies outside the measured range.

The linear pattern might continue, but we have less evidence that it will.

Therefore extrapolated predictions should be treated cautiously.

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Comparing Linear Relationships

Linear functions are especially useful when comparing options.

Suppose two phone plans are:

Plan A: C = 20 + 5x

Plan B: C = 50 + 2x

where x represents units of usage.

Plan A has:

• lower initial cost
• higher rate

Plan B has:

• higher initial cost
• lower rate

Which plan is cheaper depends on the amount of usage.

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Finding the Break-Even Point

At the break-even point:

Plan A cost = Plan B cost

Therefore:

20 + 5x = 50 + 2x

Subtract 2x:

20 + 3x = 50

Subtract 20:

3x = 30

x = 10

Find the cost:

C = 20 + 5(10)

C = 70

Therefore the plans cost the same at:

10 units of usage

with a cost of:

$70

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Interpreting the Intersection

The point:

(10, 70)

means:

At 10 units of usage, both plans cost $70.

Before the intersection, one plan is cheaper.

After the intersection, the other may become cheaper.

An intersection therefore has an important practical meaning.

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Which Plan Is Cheaper Before the Intersection?

Test:

x = 5

Plan A:

C = 20 + 5(5)

C = 45

Plan B:

C = 50 + 2(5)

C = 60

Therefore:

Plan A is cheaper at 5 units.

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Which Plan Is Cheaper After the Intersection?

Test:

x = 20

Plan A:

C = 20 + 5(20)

C = 120

Plan B:

C = 50 + 2(20)

C = 90

Therefore:

Plan B is cheaper at 20 units.

The lower rate eventually compensates for Plan B's larger starting cost.

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Comparing Slopes

Suppose:

A = 4x + 20

B = 7x + 5

The slopes are:

4 and 7

Therefore B increases faster.

This does not automatically mean B is always larger.

Its intercept is smaller.

Both slope and intercept must be considered.

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Comparing Initial Values

Consider:

A = 8x + 10

B = 5x + 40

A begins at:

10

B begins at:

40

So B starts higher.

But A has the greater rate of increase.

Eventually A may catch up and pass B.

This is why intersections are useful when comparing linear relationships.

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Wages Example

Job A pays:

A = 18h

Job B pays:

B = 15h + 60

where h is hours worked.

Job B includes an additional $60 payment but has a lower hourly rate.

Find when earnings are equal:

18h = 15h + 60

3h = 60

h = 20

At 20 hours:

A = 18(20) = 360

Both jobs pay:

$360

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Interpreting the Wage Comparison

Before 20 hours:

Job B pays more

because of the $60 initial payment.

After 20 hours:

Job A pays more

because its hourly rate is higher.

The intersection helps identify when one option becomes better than another.

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Taxi Comparison

Taxi A:

C = 3d + 4

Taxi B:

C = 2d + 10

where d is distance in kilometres.

Find when they cost the same:

3d + 4 = 2d + 10

d = 6

Cost:

C = 3(6) + 4

C = 22

At:

6 km

both taxis cost:

$22

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Interpreting a Graph with Two Lines

If two cost graphs intersect:

• the intersection represents equal cost
• below the intersection, the lower line represents the cheaper option
• above the intersection, the lower line still represents the cheaper option, although which line is lower may have changed

Graphs can therefore help with decision-making.

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Linear Models in Business

Suppose a company has:

C = 500 + 8x

where:

• C = production cost
• x = number of products

The intercept:

500

represents fixed costs.

The slope:

8

represents an additional:

$8 per product

If revenue is:

R = 13x

the company can compare cost and revenue.

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Break-Even in Business

At break-even:

Revenue = Cost

Therefore:

13x = 500 + 8x

5x = 500

x = 100

At:

100 products

revenue equals cost.

For production above 100 products, this simple model predicts that revenue exceeds cost.

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Profit as a Linear Function

Profit can be calculated as:

Profit = Revenue − Cost

Using:

R = 13x

and:

C = 500 + 8x

we get:

P = 13x − (500 + 8x)

P = 5x − 500

At:

x = 100

P = 0

This confirms the break-even point.

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Linear Models in Science

Linear functions frequently appear in science.

Examples include:

d = vt + d₀

for constant-speed motion.

V = V₀ + rt

for a container filling at constant rate.

T = T₀ + rt

for approximately constant temperature change.

The same mathematical structure appears in many different contexts.

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Motion Example

Suppose:

d = 12t + 5

where d is metres and t is seconds.

Interpret:

12 m/s

as the rate of change of distance.

Interpret:

5 m

as the starting position.

At 8 seconds:

d = 12(8) + 5

d = 101 m

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Temperature Example

Suppose:

T = −2.5t + 80

where T is °C and t is minutes.

Slope:

−2.5°C/min

Initial temperature:

80°C

After 12 minutes:

T = −2.5(12) + 80

T = 50°C

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When Is a Linear Model Appropriate?

A linear model is appropriate when:

• the rate of change is constant or approximately constant
• the data points form approximately a straight-line pattern
• the model makes reasonable predictions over the relevant range
• there is a logical reason to expect approximately constant change

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Checking Constant Rate from a Table

Consider:

Calculate:

Δy/Δx = 6/2 = 3

for each interval.

The rate remains constant.

Therefore a linear model is appropriate.

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When a Linear Model Is Not Appropriate

Consider:

The changes are:

+2, +4, +8, +16

The rate of change is not constant.

This pattern is not linear.

A linear model would not accurately represent the relationship.

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Curved Graphs Are Usually Not Linear

A linear function produces a straight-line graph.

If data consistently follow a curve, another model may be more appropriate.

Common nonlinear patterns include:

• exponential growth
• exponential decay
• quadratic relationships
• inverse relationships
• periodic relationships

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Real Data Are Not Always Perfect

Real measurements often contain variation.

Consider:

The increases are not exactly identical.

However, the data increase by approximately:

2°C per unit time

A linear model might still be reasonable.

Real data do not have to lie perfectly on a line for a linear model to be useful.

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Line of Best Fit

When data are approximately linear, a line of best fit can be used.

The line does not need to pass through every point.

Instead, it represents the overall trend.

A linear model based on a best-fit line can be used to:

• describe the trend
• estimate the rate of change
• make predictions

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Residuals

A residual measures the difference between an observed value and the value predicted by a model.

In simple terms:

residual = actual value − predicted value

Small residuals suggest that the model predicts the data reasonably well.

Large or systematic residuals may suggest that a different model would be better.

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Example of a Prediction Error

Suppose a model predicts:

y = 42

but the measured value is:

45

Then:

residual = 45 − 42

residual = 3

The model underestimated the observed value by 3 units.

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Appropriate Does Not Mean Perfect

A model is a simplified representation of reality.

A useful linear model might not predict every measurement exactly.

Instead, ask:

• Does it capture the overall trend?
• Are the errors reasonably small?
• Is the rate approximately constant?
• Does the model make sense scientifically or practically?
• Are predictions being made within a sensible range?

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Beware of Extrapolation

Suppose population data are approximately linear between 2020 and 2025.

A prediction for 2026 might be reasonable.

A prediction for the year 2200 using the same linear model would be much less reliable.

Many factors could change over such a long period.

The farther we extrapolate, the greater the risk that the original pattern no longer applies.

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Impossible Predictions

Suppose:

T = −4t + 80

models cooling.

Eventually the model predicts:

T = −1000°C

That may be physically impossible under the conditions of the problem.

This tells us the linear model is only valid for a limited time.

Mathematical models must be interpreted using real-world knowledge.

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Population Models

Suppose:

P = 5000t + 100000

where t is years after a particular starting year.

The model assumes population increases by:

5,000 people per year

This might work over a short period.

Over decades or centuries, birth rates, migration, resources, and other factors may change.

A linear model may eventually become inappropriate.

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Comparing a Table, Graph, and Equation

A linear relationship can be represented in several ways.

Equation

Shows the rate and initial value precisely.

Table

Shows specific input-output pairs.

Graph

Shows the overall pattern visually.

Verbal Description

Explains what the quantities mean in context.

Strong mathematical understanding means moving between all four representations.

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From Equation to Table

For:

y = 3x + 2

we can calculate:

The table confirms a constant increase of:

3

for every increase of:

1

in x.

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From Table to Context

Suppose:

The initial cost is:

$10

The rate is:

$4/hour

Therefore:

C = 4h + 10

A possible interpretation is a rental service charging $10 initially plus $4 per hour.

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From Context to Graph

Suppose:

C = 4h + 10

The graph begins at:

(0, 10)

and rises:

4 units vertically for each 1 unit horizontally

Every point on the graph represents a possible combination of time and cost within the meaningful domain.

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Discrete Versus Continuous Situations

Some linear relationships are continuous.

Examples:

• time
• distance
• temperature
• volume

Values between measured points can make sense.

Other situations are discrete.

Examples:

• number of students
• number of cars
• number of tickets
• number of products

If x represents tickets, values such as:

x = 3.7 tickets

usually do not make sense.

The context affects which points on a linear graph are meaningful.

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Recognizing Unreasonable Answers

Suppose a model predicts:

−8 students

This is mathematically possible as an output of an equation but not reasonable in the real situation.

Suppose a model predicts:

2.5 hours

That may be completely reasonable.

Always ask:

What does this number represent?

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Comparing Models from Graphs

Suppose two lines represent savings accounts.

Account A has:

• lower intercept
• steeper slope

Account B has:

• higher intercept
• flatter slope

This means:

• B starts with more money
• A grows faster
• A may eventually overtake B

The intersection represents the time when both accounts contain the same amount.

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Comparing Models Without Solving

Suppose:

A = 8x + 100

B = 5x + 100

Both start at the same value:

100

But:

8 > 5

Therefore, for positive x:

A grows faster than B.

We can make this comparison directly from the slopes.

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Same Slope, Different Intercepts

Suppose:

A = 4x + 20

B = 4x + 50

Both increase at the same rate.

B always remains:

30 units greater

than A.

Their graphs are parallel.

They never intersect.

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A Full Application Example

A fitness centre offers two payment plans.

Plan A:

A = 12v + 20

Plan B:

B = 8v + 52

where v is the number of visits.

Interpret Plan A

Initial fee:

$20

Cost per visit:

$12

Interpret Plan B

Initial fee:

$52

Cost per visit:

$8

Find when the costs are equal

12v + 20 = 8v + 52

4v = 32

v = 8

Cost:

A = 12(8) + 20

A = 116

Therefore:

Both plans cost $116 at 8 visits.

For fewer than 8 visits, Plan A is cheaper.

For more than 8 visits, Plan B is cheaper.

This is a complete interpretation of the two linear models.

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Evaluating a Model

Before trusting a linear model, ask:

Is the rate approximately constant?

Does the graph look approximately straight?

Does the model fit the known data reasonably well?

Are the variables logically related in a linear way?

Is the prediction within or far beyond the observed range?

Does the prediction make sense in the real world?

A mathematical equation can be calculated correctly while still being a poor model of reality.

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Common Mistakes

Mistake: Treating slope as only a number.

In applications, slope should be interpreted with units.

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Mistake: Confusing slope and intercept.

Slope is the rate of change.

Intercept is the value when the input is zero.

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Mistake: Ignoring a negative slope.

A negative slope represents a decreasing quantity.

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Mistake: Giving only a numerical answer.

Explain what the answer means in context.

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Mistake: Assuming every relationship is linear.

Check whether the rate of change is constant or approximately constant.

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Mistake: Assuming real data must be perfectly linear.

Real measurements often contain variation.

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Mistake: Extrapolating too far.

A relationship that is approximately linear over one range may not remain linear indefinitely.

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Mistake: Ignoring the meaningful domain.

A mathematical model may produce values that are impossible in the real situation.

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Mistake: Assuming the steeper line is always larger.

The intercept also matters.

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Mistake: Comparing only initial values.

A model starting lower may eventually become larger if it has a greater slope.

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Mistake: Ignoring intersections.

An intersection often represents an important event such as equal cost, equal distance, or break-even.

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Mistake: Treating discrete quantities as continuous.

Not every decimal value makes sense for counts such as people or tickets.

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Key Terms

Linear function — function with a constant rate of change and a straight-line graph.

Linear model — linear function used to represent a real-world relationship.

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

Rate of change — amount one quantity changes compared with another.

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

Initial value — starting value represented by the intercept.

Prediction — value estimated using a mathematical model.

Interpolation — prediction made within the range of known data.

Extrapolation — prediction made outside the range of known data.

Intersection — point where two graphs meet.

Break-even point — point where two financial quantities, commonly cost and revenue, are equal.

Domain — possible or meaningful input values.

Range — possible or meaningful output values.

Continuous data — data that can take values throughout an interval.

Discrete data — data consisting of separate countable values.

Line of best fit — straight line representing the overall trend of approximately linear data.

Residual — difference between an observed value and a model's predicted value.

Mathematical model — mathematical representation used to describe or predict a real situation.

Constant rate — rate of change that remains the same.

Reasonableness — whether a mathematical result makes sense in its context.

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Key Takeaways

• Linear functions are useful for situations involving constant or approximately constant rates of change.
• A common linear model is y = mx + b.
• The slope represents the rate of change.
• The y-intercept represents the initial value.
• Slope should be interpreted with appropriate units.
• Linear models can be used to calculate unknown values.
• Substituting an input allows us to predict an output.
• Setting the output to a known value allows us to solve for the input.
• Graphs provide visual information about initial values and rates of change.
• A steeper slope represents a larger magnitude of rate of change.
• Negative slopes represent decreasing relationships.
• Intersections are important when comparing two linear models.
• An intersection means both relationships have the same output for the same input.
• In financial applications, an intersection may represent equal cost or a break-even point.
• Both slope and intercept should be considered when comparing linear relationships.
• A larger slope does not necessarily mean a function is always larger.
• Interpolation predicts within the known data range.
• Extrapolation predicts beyond the known data range.
• Extrapolated predictions are generally less reliable.
• A linear model is appropriate when the rate of change is constant or approximately constant.
• Real data do not need to lie perfectly on a straight line for a linear model to be useful.
• A curved pattern may indicate that a nonlinear model is more appropriate.
• Real-world context determines the meaningful domain of a model.
• Some variables are continuous while others are discrete.
• A mathematically correct prediction can still be physically or practically unreasonable.
• Models are simplified representations and should always be evaluated rather than accepted automatically.

Check Your Understanding

1. What does the slope represent in a real-world linear model?
2. What does the y-intercept usually represent?
3. A rental cost is modeled by C = 7h + 15. Interpret both 7 and 15.
4. Use C = 7h + 15 to predict the cost of an 8-hour rental.
5. Using the same model, determine how many hours correspond to a cost of $85.
6. A tank is modeled by V = −20t + 400. Interpret the slope and intercept.
7. Determine when the tank in Question 6 becomes empty.
8. State a reasonable domain for the model in Question 6.
9. A car's position is modeled by d = 65t + 25. What does each number represent?
10. A savings account follows S = 30w + 150. Predict the savings after 12 weeks.
11. Plan A is C = 25 + 4x and Plan B is C = 55 + 2x. Find the value of x at which the plans cost the same.
12. Determine which plan in Question 11 is cheaper before and after the intersection.
13. Job A pays A = 20h and Job B pays B = 16h + 80. Determine when the earnings are equal and explain what the intersection means.
14. Explain the difference between interpolation and extrapolation.
15. A model is based on data collected for x-values between 5 and 20. Would a prediction at x = 12 be interpolation or extrapolation? What about x = 100?
16. Explain why a perfectly constant rate of change is not always required for a linear model to be useful with real experimental data.
17. The values 2, 4, 8, 16, 32 occur as x increases by 1. Explain why a linear model is not appropriate.
18. A population model predicts −5,000 people far into the future. What does this suggest about the model?
19. Two companies have cost models A = 6x + 40 and B = 3x + 100. Compare their initial costs and rates, find their intersection, and explain which company is cheaper on either side of the intersection.
20. Choose a real-world situation that could reasonably be modeled with a linear function. Define the variables, write or propose an equation, interpret its slope and intercept, make one prediction, and explain one condition under which the model would stop being appropriate.