Linear Models

What are the parameters of a linear model?

A linear model is of the form $f (x) = m x + c$
The m represents the rate of change of the function
- This is the amount the function increases/decreases when x increases by 1
  - If the function is increasing m is positive
  - If the function is decreasing m is negative
- When the model is represented as a graph this is the gradient of the line
The c represents the value of the function when x = 0
- This is the value of the function when the independent variable is not present
- This is usually referred to as the initial value
- When the model is represented as a graph this is the y-intercept of the line

What can be modelled as a linear model?

If the graph of the data resembles a straight line
Anything with a constant rate of change
- C(d) is the taxi charge for a journey of d km
- B(m) is the monthly mobile phone bill when m minutes have been used
- R(d) is the rental fee for a car used for d days
- d(t) is the distance travelled by a car moving at a constant speed for t seconds

What are possible limitations of a linear model?

Linear models continuously increase (or decrease) at the same rate
- In real-life this might not be the case
- The function might reach a maximum (or minimum)
If the value of m is negative then for some inputs the function will predict negative values
- In some real-life situations negative values will not make sense
- To overcome this you can decide on an appropriate domain so that the outputs are never negative

Exam Tip

Make sure that you are equally confident in working with linear models both algebraically and graphically as it may be easier using one method over the other when tackling a particular exam question

Worked example

The total cost, $C$ , in New Zealand dollars (NZD), of a premium gym membership at FitFirst can be modelled by the function

$C = 14.95 t + 30, t \geq 0$

where $t$ is the time in weeks.

Calculate the cost of the gym membership for 20 weeks.

2-3-1-ib-ai-sl-linear-models-a-we-solution

Find the number of weeks it takes for the total cost to exceed 1500 NZD.

2-3-1-ib-ai-sl-linear-models-b-we-solution

Under new management, FitFirst changes the initial payment to 20 NZD and the weekly cost to 19.25 NZD. Write the new cost function after these changes have been.

2-3-1-ib-ai-sl-linear-models-c-we-solution

Linear Models (DP IB Maths: AI HL)

Revision Note