# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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Example. Draw an arrow graph for the relation “is greater than” between the set of numbers {2, 4, 6} and another set {2, 4, 5}, write down the ordered pairs for this relation, and draw a Cartesian graph.

ASYMMETRY 43

Solution. The arrow graph is drawn in figure b. The ordered pairs are {(4, 2) (6, 2) (6, 4) (6, 5)}.

The Cartesian graph is shown in figure c.

The set of numbers {4,6}, which includes the first numbers in the ordered pairs, is called the domain. The set of numbers {2,4, 5}, which includes the second numbers in the ordered pairs, is called the range.

References: Cartesian Coordinates, Domain, Image, Inverse Relations, Ordered Pairs, Range.

ARROWHEAD

Reference: Kite.

ASSOCIATIVE LAW

References: Addend, Commutative Law, Distributive Law.

ASYMMETRY

Reference: Symmetry.

44 ASYMPTOTE

ASYMPTOTE

A straight line is an asymptote of a curve if the line and the curve get closer and closer together, but never actually meet. In figure a, the straight line is an asymptote of the curve. In this example as x gets larger and larger, the curve and the asymptote get closer and closer together. The distance between them is forever diminishing.

Example. B ill plans to build a chicken run in the shape of a rectangle that will have an area of 96 square meters. Not being sure about its dimensions, he calls the length of the run x meters and the width y meters. He writes down a few possible values for x and y. For example, when the length isx = 12 meters the width is y = 8 meters, so that the area is 96 square meters. Being a well-organized person, he enters the various values of x and y into a table in order to analyze them.

x (length) in meters 3 4 6 8 12 16 24 32

y (width) in meters 32 24 16 12 8 6 4 3

Figure b is the graph of the values from the table for the length and width in meters of the chicken run. This graph is called a rectangular hyperbola.

Bill soon realizes that as x gets bigger, y gets smaller, and the bigger x gets, the smaller y gets. In fact, as x gets bigger and bigger, y gets closer and closer to

AVERAGE SPEED 45

zero. For example, when x = 1000, y = 0.096, but of course in practical terms Bill would not use these values as dimensions of his chicken run. The curve and the x-axis get closer and closer together as x gets larger. This means that the x-axis is the horizontal asymptote to the curve. Similarly, the y-axis is the vertical asymptote to the curve.

Reference: Exponential Curve, Hyperbola, Logarithmic Curve, Rectangular Hyperbola, Table of Values, Tangent.

AVERAGE

This is a quantity that refers to the three different statistical terms mode, median, and mean. The mean is also known as the arithmetic mean, which is sometimes incorrectly called the average.

Reference: Mean, Median of a Set of Data, Mode.

AVERAGE SPEED

The average speed for a journey is calculated by dividing the total distance traveled to complete the journey by the total time taken for the whole journey. If there are any stops during the journey, they are usually included in the total time taken for the journey. The formula for average speed is

total distance traveled

Average speed =--------------------------

total time taken

Care must be taken to ensure that the units of distance and time are compatible. For an average speed that is measured in kilometers per hour (km/h), the distance must be in kilometers and the time in hours. Some other units of speed in general use are as follows:

♦ Centimeters per second, cm s-1 or cm/s

♦ Meters per second, m s-1 or m/s

♦ Miles per hour, mi hr1 or mi/h

Example. John drives his family out into the countryside for afternoon tea at Madge’s Tearoom. The outward journey from home of 75 km takes them l| h. Tea with Madge takes | h, and the family return home by the same route. This return journey takes l| h. Find his average speed for the whole trip.

46 AXIS OF SYMMETRY

Solution. Write

The total distance traveled = 75 + 75

150 km

The total time taken = 1 —|-----------|- 1 -

4 2 2

4

Using the formula for average speed

46.2 (to 1 dp)

The average speed for the whole trip is 46.2 km/h.

Note. Suppose a journey was in two parts of equal distances. The speed for the first part was 30 km/h and the speed for the second part was 40 km/h. If we calculated the mean of these two speeds it would yield a different answer from the average speed. The mean speed is (30 + 40) 2 = 35 km/h. Check in the entry Algebraic Fractions

for the correct answer to the average speed for this problem.

References: Average, Gradient, Mean.

Reference: Theorem.

AXIS OF SYMMETRY

An axis of symmetry is a straight line that divides a shape into two identical halves that are mirror images of each other. An axis of symmetry is often labeled “m” on a figure (see figure a).

A point P on one side of the axis of symmetry has an image point P; on the other side, and P and P; are equidistant from the axis of symmetry. This is true for other points and their images like Q and Q '. The straight line joining P to P; is at right angles

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