Linear Models (DP IB Applications & Interpretation (AI)): Revision Note

Linear Models

What are the parameters of a linear model?

• A linear model is of the form 

• 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