Topics
3. Linear Graphs and Models
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AustraliaVIC
VCE 11 General 2023

3.05 Linear models

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Lesson

Introduction

The following has been covered so far in this chapter:

  •  how to identify linear graphs from tables of values 

  •  the slope and axis intercepts of a linear graph 

  •  finding the equation of a linear function  , and

  •  sketching linear functions from equations and other given information 

These techniques can now be used to solve a range of real life applications. It's all the same mathematics, but this time it will be applied to a given context. Applying linear functions to real life applications is known as linear modelling.

Linear models

When modelling any real-life scenario with a linear function, a range of values must be considered that makes sense for that situation. In mathematics, this is called the domain.

For instance a helicopter flies for 4 hours at a constant speed of 315 km/h before reaching its destination. If D represents the distance in kilometres, and t represents the time elapsed in hours, what is the domain?

This model will stop being valid when the helicopter stops flying, so the domain will only include t-values for when the helicopter is flying, where t is the time in hours.

The domain for this linear model would be written as 0\leq t \leq 4. To interpret this domain, we read it as, the values of t start at 0 hours and end at 4 hours.

Examples

Example 1

The amount of medication M (in milligrams) in a patient’s body gradually decreases over time t (in hours) according to the equation M=1050 - 15t.

a

After 61 hours, how many milligrams of medication are left in the body?

Worked Solution
Create a strategy

Substitute t=61 into the given equation.

Apply the idea
\displaystyle M\displaystyle =\displaystyle 1050 - 15tWrite the equation
\displaystyle =\displaystyle 1050 - 15 \times 61Substitute t=61
\displaystyle =\displaystyle 1050 - 915Evaluate the multiplication
\displaystyle =\displaystyle 135 \text{ mg}Evaluate
b

How many hours will it take for the medication to be completely removed from the body?

Worked Solution
Create a strategy

Substitute M=0 into the given equation.

Apply the idea
\displaystyle M\displaystyle =\displaystyle 1050 - 15tWrite the equation
\displaystyle 0\displaystyle =\displaystyle 1050 - 15tSubstitute M=0
\displaystyle 15t\displaystyle =\displaystyle 1050Add 15t to both sides
\displaystyle t\displaystyle =\displaystyle 70 \text{ hours}Divide both sides by 15

Example 2

Deborah left for a road trip at midday. The following graph shows the total distance travelled (in kilometres), t hours after midday:

2
4
6
8
hours
110\text{ km}
220\text{ km}
330\text{ km}
440\text{ km}
550\text{ km}
660\text{ km}
770\text{ km}
880\text{ km}
990\text{ km}
\text{distance travelled (km)}
a

Find the slope of the straight line.

Worked Solution
Create a strategy

Use the formula: \text{slope}= \dfrac{y_2 - y_1}{x_2 - x_1}

Apply the idea

By choosing two points that lie on the line we can let (x_1,y_1)=(0,0) and (x_2,y_2)=(2,220).

\displaystyle m\displaystyle =\displaystyle \frac{220 - 0}{2 - 0}Substitute the coordinates
\displaystyle =\displaystyle 110Evaluate

This means that for every 1 hour the car travels 110 kilometers.

b

What does the slope of the line represent?

A
The total distance travelled
B
The car’s speed
C
The car's acceleration
D
The slope of the road
Worked Solution
Create a strategy

The slope represents the rate of change of the distance travelled for a change in time.

Apply the idea

From the graph, we can see that the distance travelled is increasing as the time increases. So the slope represents the car's speed, option B.

Example 3

The graph shows the amount of water remaining in a bucket that was initially full before a hole was made in its side.

4
8
12
16
20
24
28
32
36
\text{time (minutes)}
4
8
12
16
20
24
28
32
36
\text{quantity (litres)}
a

What is the slope of the function?

Worked Solution
Create a strategy

Use the formula: \text{slope}= \dfrac{y_2 - y_1}{x_2 - x_1}

Apply the idea

By choosing two points that lie on the line we can let (x_1,y_1)=(0,34) and (x_2,y_2)=(12,28).

\displaystyle m\displaystyle =\displaystyle \frac{28 - 34}{12 - 0}Substitute the coordinates
\displaystyle =\displaystyle -\dfrac{1}{2}Evaluate
Reflect and check

In context, this means that for every 1 minute that passes the bucket loses \dfrac{1}{2} a litre of water.

b

What is the y-value of the y-intercept?

Worked Solution
Create a strategy

The y-intercept is the value of at which the line crosses the y-axis or when x=0.

Apply the idea

Looking at the graph, the y-value at which the line crosses the y-axis is 34.

Reflect and check

In context, this means that initially (when x=0) there was 34 litres of water in the bucket.

c

Write an equation to represent the amount of water remaining in the bucket, y, as a function of time, x.

Worked Solution
Create a strategy

Use the form y = mx + c

Apply the idea

From parts (a) and (b), we found that m = -\dfrac{1}{2} and c = 34.

\displaystyle y\displaystyle =\displaystyle -\dfrac{1}{2} \times x + 34Substitute m and c
\displaystyle y\displaystyle =\displaystyle -\dfrac{1}{2}x + 34Simplify
Idea summary

In linear modelling situations:

  • The y-intercept represents the initial value.

  • The slope represents the rate of change.

Outcomes

U1.AoS4.2

the concept of a linear model and its properties, and simultaneous linear equations and their solutions

U1.AoS4.5

develop a linear model to represent and analyse a practical situation and specify its domain of application

U1.AoS4.6

interpret the slope and the intercept of a straight-line graph in terms of its context and use the equation to make predictions

U1.AoS4.7

construct graphs and/or tables of values for given linear models and formula and vice versa

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