1.7 Linearisation Using Exponentials

Note: if you cannot remember the appearance of graphs of the form y=kx^{n}, revise notes for 1.6 Exponential Graphs.

Linearisation using x^{-1}

  • If a scatterplot appears to show a relationship of the form y=kx^{-1}, it can be linearised by taking the inverse of the data points.

Example

X

Y

1

1

2

0.5

3

0.29

4

0.25

5

0.21

6

0.17

7

0.14

A company running a streaming service records the proportion of first-time users which return to the service over a period of 7 months, the results are plotted above. They require the data to be linearised in order to perform further analysis. From the dot plot, we can see that the relationship appears to be of the form: y=kx^{-1} and so we will take the inverse of the x data in order to linearise it:

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1.6 Exponential Graphs

Graphs of y=kxn

  • In Further Maths, we will analyse graphs of the form y=kx^{n}, where k is a constant known as the constant of proportionality, for the following values of n: -2, -1, 1, 2, 3. These forms are detailed individually below.

n= -2

  • The graph of y=kx^{-2} is symmetrical about the line x=0 and has two asymptotes. One at x=0, and one at y=0.
  • At x=0, the graph is undefined and on either side the graph curves upwards and goes towards infinity.
  • As you move away from the origin, the graph curves inwards and the value of y gets closer and closer to 0 the further you are from the origin.
  • All values of y are positive if k is positive, or negative if k is negative.

Example

The graph of y=3x^{-2} is shown below:

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3.2 Modelling Linear Associations

Identifying Explanatory and Response Variables

  • It is important to correctly select the explanatory and response variables when using regression, or the relationship will be incorrect.
  • The explanatory variable is the variable which is used to explain or predict the response variable.
  • In a conventional x-y dataset, the x variable is the explanatory variable and y is the response variable.

Fitting Least Squares Models

  • Start by identifying the explanatory and response variables.
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2.5 Relationships between two Numerical Variables

Guidelines to Analysing Numerical Associations

  • Begin with context: what does the data represent?
  • Identify the explanatory and response variables.
  • Assess the form of the association: is it linear, non-linear or is there no association.
  • If it is linear, assess the strength (strong, moderate or weak). Ideally, do this using the Pearson’s correlation coefficient (detailed in 2.6 Pearson’s Correlation Coefficient), however if the raw data is not available, a qualitative assessment will suffice.
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