Level 3MathsAQAMathematical Studies (Core Maths)Paper 2C: Graphical TechniquesRevision NotesGraphical MethodsGraphical MethodsModelling with Graphs

Modelling with Graphs (AQA Level 3 Mathematical Studies (Core Maths)): Revision Note

Exam code: 1350

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

Modelling with Graphs

How are graphs used to model real situations?

  • Real scenarios can be modelled using graphs

    • These can then be used to help solve problems

  • Different scenarios will require different types of graph

    • This includes the previously covered linear, quadratic, cubic, and exponential graphs

  • A common use of these graphs is to estimate data points that are not known

    • E.g. Start by measuring 10 data points for the real life scenario

    • Plot an appropriate graph

    • This graph can then be used to interpolate values that were not directly measured from the scenario

  • Interpolation is the process of reading data points from within the range of known data

    • E.g. If data is known between x equals 2 and x equals 10, data for x equals 5 can be found by interpolation

  • Extrapolation is the process of reading data points from beyond the range of known data

    • E.g. If data is known between x equals 2 and x equals 10, data for x equals 12 can be found by extrapolation

  • Extrapolation is much less valid than interpolation

    • This is because there is no evidence for if the trend or relationship continues outside of the collected data range

Worked Example

Jodie is tracking the value of her new car to model its depreciation.

She collects the following data using online valuations for her car for the first 5 years. The values are rounded to the nearest hundred pounds.

Number of years, t

Value in £, V

0

20 000

1

17 600

2

15 600

3

13 700

4

12 100

5

10 700

(a) Plot a graph of this data and join the points with a smooth curve.

Answer:

Plot the values as accurately as you can and then join with a smooth curve

A downward curve with points plotted, showing the value of a car depreciating

(b) Suggest two types of model which may fit this data.

Answer:

Consider the shape of the graph
It is not a straight line, so it is not a linear graph
It is decreasing either towards a stationary point, which suggests it could be quadratic
Or it could be continually decreasing, tending towards a lower limit, which suggests an exponential model

Quadratic or Exponential

(c) Jodie chooses to model the data with an equation: V equals 120 t squared plus b t plus c.

Using the data in the table, find possible values for b and c.

Answer:

The value of c is the y-intercept, when t equals 0
This can be seen in the table and on the graph, as 20 000

bold italic c bold equals bold 20 bold space bold 000

The model is now

V equals 120 t squared plus b t plus 20 space 000

Use one of the data points from the table to substitute into the equation
Using the last value in the table, V equals 10 space 700 and t equals 5

10 space 700 space equals space 120 open parentheses 5 squared close parentheses plus b open parentheses 5 close parentheses plus 20 space 000

Solve for b

table row cell negative 9300 end cell equals cell 120 open parentheses 5 squared close parentheses plus 5 b end cell row cell negative 9300 space end cell equals cell space 3000 space plus space 5 b end cell row cell negative 12 space 300 space end cell equals cell space 5 b end cell end table

bold italic b bold equals bold minus bold 2460

Note that selecting a different data point may result in a different answer for b, which is why the question asks for "possible" values - it is an approximate model, not an exact formula

(d) Suggest why this equation may not be suitable to model the value of the car.

Answer:

The suggested model is bold italic V bold equals bold 120 bold italic t to the power of bold 2 bold minus bold 2460 bold italic t bold plus bold 20 bold space bold 000 which is a positive quadratic.

The downward part of the quadratic will model the decrease in value well, but a positive quadratic will also have an upward curve, which will not model the decline in value accurately at all.

Using this positive quadratic model would predict that the car would start increasing in value, at an increasing rate, after the minimum value, which is highly likely to be incorrect.

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths Content Creator

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    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.