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DP IB Maths: AI SL

Revision Notes

Home / IB / Maths: AI SL / DP / Revision Notes / 2. Functions / 2.3 Modelling with Functions / 2.3.1 Linear & Piecewise Models


2.3.1 Linear & Piecewise Models


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

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 m­i
    • The value if the independent variable was not present ci

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?

  • Linear models have a constant rate of change
    • In real-life this might not be the case
    • A function might increase (or decrease) gradually rather than at a constant rate

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



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Exponents & Logarithms
          • 1.1.3 Approximation & Estimation
            • 1.1.4 GDC: Solving Equations
            • 1.2 Sequences & Series
              • 1.2.1 Language of Sequences & Series
                • 1.2.2 Arithmetic Sequences & Series
                  • 1.2.3 Geometric Sequences & Series
                    • 1.2.4 Applications of Sequences & Series
                    • 1.3 Financial Applications
                      • 1.3.1 Compound Interest & Depreciation
                        • 1.3.2 Amortisation & Annuities
                      • 2. Functions
                        • 2.1 Linear Functions & Graphs
                          • 2.1.1 Equations of a Straight Line
                          • 2.2 Further Functions & Graphs
                            • 2.2.1 Functions
                              • 2.2.2 Graphing Functions
                                • 2.2.3 Properties of Graphs
                                • 2.3 Modelling with Functions
                                  • 2.3.1 Linear & Piecewise Models
                                    • 2.3.2 Quadratic & Cubic Models
                                      • 2.3.3 Exponential Models
                                        • 2.3.4 Direct & Inverse Variation
                                          • 2.3.5 Sinusoidal Models
                                            • 2.3.6 Strategy for Modelling Functions
                                          • 3. Geometry & Trigonometry
                                            • 3.1 Geometry Toolkit
                                              • 3.1.1 Coordinate Geometry
                                                • 3.1.2 Arcs & Sectors
                                                • 3.2 Geometry of 3D Shapes
                                                  • 3.2.1 3D Coordinate Geometry
                                                    • 3.2.2 Volume & Surface Area
                                                    • 3.3 Trigonometry
                                                      • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                        • 3.3.2 Non Right-Angled Trigonometry
                                                          • 3.3.3 Applications of Trigonometry & Pythagoras
                                                          • 3.4 Voronoi Diagrams
                                                            • 3.4.1 Voronoi Diagrams
                                                              • 3.4.2 Toxic Waste Dump Problem
                                                            • 4. Statistics & Probability
                                                              • 4.1 Statistics Toolkit
                                                                • 4.1.1 Sampling & Data Collection
                                                                  • 4.1.2 Statistical Measures
                                                                    • 4.1.3 Frequency Tables
                                                                      • 4.1.4 Linear Transformations of Data
                                                                        • 4.1.5 Outliers
                                                                          • 4.1.6 Univariate Data
                                                                            • 4.1.7 Interpreting Data
                                                                            • 4.2 Correlation & Regression
                                                                              • 4.2.1 Bivariate data
                                                                                • 4.2.2 Correlation Coefficients
                                                                                  • 4.2.3 Linear Regression
                                                                                  • 4.3 Probability
                                                                                    • 4.3.1 Probability & Types of Events
                                                                                      • 4.3.2 Conditional Probability
                                                                                        • 4.3.3 Sample Space Diagrams
                                                                                        • 4.4 Probability Distributions
                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                            • 4.4.2 Expected Values
                                                                                            • 4.5 Binomial Distribution
                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                • 4.6 Normal Distribution
                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                    • 4.7 Hypothesis Testing
                                                                                                      • 4.7.1 Hypothesis Testing
                                                                                                        • 4.7.2 Chi-squared Test for Independence
                                                                                                          • 4.7.3 Goodness of Fit Test
                                                                                                            • 4.7.4 The t-test
                                                                                                          • 5. Calculus
                                                                                                            • 5.1 Differentiation
                                                                                                              • 5.1.1 Introduction to Differentiation
                                                                                                                • 5.1.2 Applications of Differentiation
                                                                                                                  • 5.1.3 Modelling with Differentiation
                                                                                                                  • 5.2 Integration
                                                                                                                    • 5.2.1 Trapezoid Rule: Numerical Integration
                                                                                                                      • 5.2.2 Introduction to Integration
                                                                                                                        • 5.2.3 Applications of Integration
                                                                                                                      Daniel Finlay

                                                                                                                      Author: Daniel

                                                                                                                      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.


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