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FM Construction of Graphs

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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1.5 Use of Data and Graphs

Constructing Graphs from Tables

  • Given a table containing the x and y values of several points on a graph, we can construct a graph by plotting the known points and draw between them, using the trend of points to guess the overall shape of the graph.
  • When guessing the shape of the graph, there are many possible options. It is best to go with the simplest option.

Example

X

Y

0.5

11

1

8.5

1.5

5.5

2

4

3

2

4.5

1.8

6

1.2

9

1

We wish to construct a graph from the above table. To begin, we plot the points on a graph:

Picture 2

Now, we draw between these points. In this case, the overall trend appears to curve and so we will draw a curved line:

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1.4 Analysing Non-Linear Graphs

Intercepts of Non-Linear Graphs

  • Non-linear graphs can have no intercepts, one intercept or multiple intercepts along either axis.

Example

Picture 2

Above is a graph showing the profits earned by a community theatre in the years before they changed management. In the context of this situation, we can say that the intercepts at t=-3, -2 and 1 years represent points at which the community theatre just broke even (i.e. no profit or loss). The intercept at P=$1000 represents the profits made in the year the new management took over.

Maximum/Minimum Points

  • It may interest us to know the highest, or lowest values that the y-variable has taken. For example, if we want to find the highest a tide has been in a year in order to know how high to build a support structure.
  • The maximum and minimum points of non-linear graphs will be points of 0 gradient or the endpoints of a section (i.e. where a graph ends or undergoes structural change).
  • There may be multiple minimum points and/or multiple maximum points.
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1.3 Simultaneous Linear Equations

Systems of Simultaneous Linear Equations

  • To solve all unknown values in a set of simultaneous linear equations, the number of unknowns must be the same as the number of unique equations in the set.

Number of Solutions for Simultaneous Linear Equations

For two simultaneous linear equations

y=m_{1} x+c_{1}

y=m_{1} x+c_{1}

The number of solutions is defined by the gradient and y-intercept of the two equations.

Unique Solution (m_{1} \neq m_{2})

        • There is a single value for each variable that satisfies every equation.

No Solutions (m_{1} = m_{2}, and c_{1} \neq c_{2})

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1.2 Segment and Step Graphs

Segment Graphs

  • Segment graphs consist of a series of straight-line graphs, creating one continuous line.
  • Segment graphs are used to model situations which change at distinct points.
  • Each segment can be analysed individually as a straight-line graph.

Example

Picture 1

Above is a segment graph modelling the distance a driver is from their house on a particular trip. We can see the first segment begins at (0,0) and ends at (1,40). We can use these points to find the gradient of the segment, which in this case, tells us the speed (km/hr) which the driver was travelling:

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1.1 Straight Line Graphs

Straight Line Graphs

  • Relationships of the form: y=mx+c, where m is the gradient and c is the y-intercept, can be modelled graphically as a straight line.

Example

Picture 1

Above is the graph for the linear relationship: y=-2x+1

Intercepts

  • Two points which are often of interest in graphs are the points at which they cross the x-axis, and where they cross the y-axis. These are known as the x-intercept and y-intercept, respectively.
  • To find the x-intercept given a formula, set y to 0 and solve for x.
  • To find the y-intercept given a formula, set x to 0 and solve for y.
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