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Ken needs to know whether or not his business is growing and why. He may want to employ more staff to keep up with sales, or lay off staff if the business is declining. The moving averages suggest that the business is growing, but at a slower rate in the third year. The spring sales have declined and the growth rates for summer and fall have slowed. But there has been a spurt in winter sales. This inconsistent pattern makes it difficult to predict the future with assurance. There is other information that may need to be examined and incorporated into the analysis, for example, the weather patterns, if it is true that people eat more ice cream in warm winters than cold winters. Has advertising had any impact, or has competition from other companies affected sales during the 3 years? Have there been any price changes? It is important not to rely solely on the trends in a time series when other influences may be at work.
References: Average, Graphs, Mean, Statistics.
Reference: Imperial System of Units.
A tonne is a unit of weight and is equal to 1000 kilograms. A builder having a truckload of sand delivered to a building site would expect several tonnes to arrive. One tonne of water has a volume of 1 cubic meter at 4°C.
References: CGS System of Units, Gram, Kilogram, Metric Units, SI Units. TRANSFORMATION GEOMETRY
Transformation geometry in this book is the study of how sets of points and their positions in space are changed by enlargements, reflections, rotations, and translations. In fact the whole of space is transformed, but we focus on specific points or shapes. The original set of points or shape is called the obj ect, and the set of points or shape that results from the transformation is called the image. Properties of the original shape that do not change after the transformation are called invariant properties. For example, the invariant properties may be area, shape, angle size, and so on. The transformations reflection, rotation, and translation leave size and shape invariant (this includes lengths of lines, the area of the shape, and angles in the shape). Transformations that do this are called isometries. It is clear that enlargement is not an isometry, because it changes the size of the shape. The following transformations are discussed in more detail under the appropriate entries: enlargement, reflection, rotation, and translation.
References: Congruent Figures, Image, Invariant Points, Object.
A translation is one of four transformations described in this book, and is defined in terms of a grid, such as the squares on a chessboard. Column vectors are used to represent translations. Figure a shows part of a chessboard, with the piece known as the queen on a white square. The queen has been chosen for this demonstration because it can move any number of squares in various directions, provided it lands
c % D
on a square. As the queen moves from one position to another its movement is a translation. The changes in the position of the queen are defined by two numbers in brackets, with one written above the other. The top number states how many squares the queen moves in a direction from side to side, and the bottom number describes how many squares it moves up or down. If the numbers are negative, the queen moves in the opposite direction.
Examples of sideways movement:
♦ 2 means two squares to the right
♦ —3 means three squares to the left
Examples of up and down movement:
♦ 5 means five squares up.
♦ — 1 means one square down.
The translation that moves the queen to square A is 2 squares to the right and 2 squares up, which is written (;?). Queen to B is written (~?). Queen to
9 2 0
C is written ( 0 ). Queen to D is written (0). Queen to E is written (_2). If the Queen does not move at all, the translation is written (®).
Translations of shapes can take place on a Cartesian plane, as illustrated by the following example. The initial position of the shape is called the object and the final position is called the image, which is the convention for all transformations. It is also the convention in transformation geometry that not only does the object change its position by the transformation, but so does the whole of the Cartesian plane. A translation has no invariant points.
Example. With reference to figure b, state the vector for each of the translations where F' is the image of F and F" is the image of F.
Solution. Every point on the flag undergoes the same translation. So we choose one point on F, say the top of the flagpole, and find its image on F'. The translation of the top of the flagpole is G*), so this is the vector for the whole flag. Similarly, the vector that translates F to F" is (_^5).
Notation We say that the flag F maps onto its image F' under the translation (), and F maps onto F” under the translation (_^5). The symbol for “maps onto” is so we write F ->■ F!.