IBMaths: AI SLDPRevision Notes2. Functions2.3 Modelling with Functions2.3.1 Linear & Piecewise Models

Linear & Piecewise Models (DP IB Maths: AI SL)

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Linear Models

What are the parameters of a linear model?

  • A linear model is of the form space f left parenthesis x right parenthesis equals m x plus 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,space C, in New Zealand dollars (NZD), of a premium gym membership at FitFirst can be modelled by the function

space C equals 14.95 t plus 30 comma blank t greater or equal than 0

where space t is the time in weeks.

a)
Calculate the cost of the gym membership for 20 weeks.

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

b)
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

c)
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

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Linear Piecewise Models

What are the parameters of a piecewise linear model?

  • A piecewise linear model is made up of multiple linear models space f subscript i open parentheses x close parentheses equals m subscript i x plus c subscript i
  • For each linear model there will be
    • The rate of change for that interval, m subscript i
    • The value if the independent variable was not present, c subscript i

What can be modelled as a piecewise linear model?

  • Piecewise linear models can be used when the rate of change of a function changes for different intervals
    • These commonly apply when there are different tariffs or levels of charges
  • Anything with a constant rate of change for set intervals
    • C(d) is the taxi charge for a journey of d km
      • The charge might double after midnight
    • R(d) is the rental fee for a car used for d days
      • The daily fee might triple if the car is rented over bank holidays
    • s(t) is the speed of a car travelling for t seconds with constant acceleration
      • The car might reach a maximum speed

What are possible limitations of a piecewise linear model?

  • Piecewise linear models have a constant rate of change (represented by a straight line) in each interval
    • In real-life this might not be the case
    • The data in some intervals might have a continuously variable rate of change (represented by a curve) rather than a constant rate
    • Or the transition from one constant rate of change to another may be gradual- i.e. a curve rather than a sudden change in gradient

Exam Tip

  • Make sure that you know how to plot a piecewise model on your GDC

Worked example

The total monthly charge, £ C, of phone bill can be modelled by the function

space C left parenthesis m right parenthesis equals open curly brackets table row cell 10 plus 0.02 m end cell row cell 9 plus 0.03 m end cell end table blank table row cell 0 less or equal than m less or equal than 100 end cell row cell m greater than 100 end cell end table close,

wherespace m is the number of minutes used.

a)
Find the total monthly charge if 80 minutes have been used.

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

b)
Given that the total monthly charge is £16.59, find the number of minutes that were used.

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

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Dan

Author: Dan

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.