When many people think of functions, they think of something likebut functions have a broader definition than expressions in x. A function is actually a one to one mapping between sets. The sets can be almost anything, but most frequently we are concerned with sets of number, and functions which numbers from one set onto […]

## Vectors I

A vector is the difference between two points. If two points areandthen we can write the difference(when written in a column between brackets) asThe arrow above means that we are going from the pointto the pointIf we swapandthen we are going fromtoand writeThis is shown below. We can also write points as pairs of coordinates […]

## Drawing a Line of Best Fit

When drawing a line of best fit it to a set of data pointsis important to realise that the line should 1. Pass through the centre of the pointswhereis the mean of the x values andis the mean of the y values. 2. Should have as many points – or as close as possible – […]

## Direct and Inverse Proportion

The two graphs show quantities directly proportional in the first graph and inversely proportional in the second graph. For the first graph we can write down, and for the second graph where, in each case,is a constant of proportionality. We are typically asked something like: The amount of petrolused by a car is directly proportional […]

## Quadratic Sequences

A quadratic sequence is generated by any rule of the form The problem is often to find the rule for a particular given sequence. An example of a quadratic sequence is: 2, 4, 8, 14, 22 When we find the difference line we obtain 2 4 8 14 22 2 4 6 8 The difference […]

## Similar Triangles Formed Between Parallel Lines

When two lines cross each other at a point A, angles on opposite sides of the point are the same. If parallel lines are drawn to construct two triangles, with one angle in each triangle the angle at A, then the triangles will be similar. In the diagram below, triangles ABC and tirangles ADE are […]

## Ratios and Similar Shapes

If a shape is enlarged, so that all the sides are multiplied by the same factor, it will remain the same basic shape, the only difference being that all the sides are larger by the same factor. Sometimes though, we may deconstruct a diagram into similar shapes so that an enlargement exists from one shape […]

## Nets and Surface Areas

To find the surface area of a solid shape, we should often first draw the net of the solid. This involves ‘unfolding’ the solid onto a flat surface and finding the areas of individual parts of the net, which are often simple shapes like rectangles, squares, triangles and circles. To find the area of the […]

## Volumes of Frustums

A frustum is that past of a pyramid or cone that lines between two parallel planes cutting it. If one of the planes is the base and the other is horizontal then the result is the shape shown below right (for a cone). The smaller cone top right and the original cone left are similar […]

## Profit and Loss

Every businessman wants to make a profit, and as large a profit as possible. This means selling your wares at a higher cost thean your bought them for, plus whatever costs have been incurred. Making a profit is not always possible though. Sometimes it is necessary to view your expenses as done deals, and your […]