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Example. Madge collected data about the weights of 30 colleagues in her office. She split the weights into class intervals of 5 kg and recorded the data in a frequency table as shown. Draw a histogram of the results.
Solution. The class interval 40- means a weight of 40-45 kg that includes 40 kg, but does not include 45 kg. The use of class intervals is common for continuous data. Madge drew a histogram of the frequency table, ensuring the axes were clearly labeled, there was no space between the columns, and the graph had a title.
Weight in kg (x) Frequency (f )
Weights of Madge’s Office Workers
50 60 70
Weight in kg
References: Bar Graph, Class Interval, Continuous Data, Frequency Distribution.
Horizontal, which is also known as level, is at right angles to the vertical. To define vertical, we can use a device called a plumb line. This is a length of string fastened at one end to a small weight called a plumb bob. The other end of the string is fixed, say to a beam, and if the small weight is allowed to hang freely without swinging, the string will be vertical (see figure). If the beam is at right angles to the string, it will be a horizontal beam.
Vertical * line
Reference: Right Angle.
Reference: Inclined Plane.
This is a curve that consists of two separate branches, which are sometimes called arms. The equation of a hyperbola is of the type y = a/x, or its equation may be written as xy = a, where a is a positive or negative number. A hyperbola has asymptotes. They are not part of the graph, but are usually drawn to accompany the graph. Some of the properties of a hyperbola are covered in the example.
Example. Use a table of values to draw the graph of the hyperbola which has the equation y = 12/x.
Solution. Suppose we use x values between —12 and 12, which will provide plenty of points to plot. We can reduce the amount of calculation by being selective in our choice of a: values. The y value corresponding to each a: value is calculated by dividing the x value into 12. The results are recorded in the table of values. When x = 0, y is undefined, because a number cannot be divided by zero.
x -12 -10 -8 -6 -4 -2 -1 0 1 2 4 6 8 10 12
y -1 -1.2 -1.5 -2 -3 -6 -12 undefined 12 6 3 2 1.5 1.2 1
From the table of values, it can be seen that as the x values get bigger and bigger, the y values get closer and closer to zero. In this way the x-axis, whose equation is
y = 0, becomes the horizontal asymptote. In a similar way, the y-axis is the vertical asymptote. More information is provided in the entry Asymptote.
The coordinates are plotted on a graph (see the figure). The graph has half-tum rotational symmetry about the origin. Since the two asymptotes are at right angles, like the sides of a rectangle, the graph is often called a rectangular hyperbola. The gradient of this curve is always negative.
The graph of y = 3/x would be similar to the graph shown and have the same asymptotes, but it would be closer to the axes. The graph of y = —<4/x would be drawn in the other two quadrants, and would be further out from the axes than y = 3/x, but closer to the axes than y = 12/x, and have the same asymptotes. The gradient of y = — 4/x is always positive.
When one quantity varies inversely as another quantity, the graph of their relationship is a hyperbola. We say that the two quantities are inversely proportional to each other. For example, suppose you are building a house and deciding how many workers to employ. The more workers you employ, the shorter is the time to complete the building of the house. Time (?) varies inversely with the number of workers (m), and the graph of t against m is a hyperbola. Of course, the graph will only be in the positive quadrant, because you cannot have a negative number of workers!
References: Asymptote, Conic Sections, Gradient, Graphs, Proportion, Quadrants, Table of Values.
In a right-angled triangle the hypotenuse is the name of the side that is opposite to the right angle. In a right-angled triangle the hypotenuse is always the longest side.
References: Pythagoras’ Theorem, Right Angle, Symmetry, Trigonometry.
This is a polyhedron that has 20 faces; the prefix icosa means 20. The regular icosahedron has 20 congruent faces that are equilateral triangles. A regular icosahedron is one of the Platonic solids. The figure shows a regular icosahedron with its net, which is made up of 20 equilateral triangles.
References: Congruent Figures, Equilateral Triangle, Face, Net, Platonic Solids, Polyhedron, Regular Polyhedron.
An image is the figure obtained after applying a transformation to a shape, called an object. The transformations studied in this book are found under the following entries: Enlargement, Reflection, Rotation, Translation.
Reference: Transformation Geometry.
IMPERIAL SYSTEM OF UNITS