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# FM Non-Linear Graph

## 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:

## 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

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

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

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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