CHAPTER IV GRAPHICAL REPRESENTATION 17. Definitions and first principles. Mathematical relations are frequently made clearer when exhibited to the eye by means of a properly constructed diagram. The method of representing mathe- matical relations by diagrams is called graphical representation. The simplest graphical representation is the representation of a single number, such as the measure of a length or of a sum of money, along a straight, line. Such a representation is accom- plished if, with an arbitrary unit of measure, we lay off from an arbitrary point in a straight line a length whose measure is the same as that of the number we seek to represent. For example, five years may be graphically represented by a line five inches long. It should be noted that the same line may just as well be used to represent five million years. If we wish to represent a magnitude capable of taking both positive and negative values, the positive values are laid off to the right of the arbitrary point and the negative values to the left. The arbitrary line is called an axis; the arbitrary point from which measurements are made is called the origin, or the zero point, and the unit employed is called the unit of measurement. Graphical representation assumes its chief importance in the representation of functions of one or more variables. For present purposes a function of one variable may be defined as a mathe- matical expression which depends for its value upon the value of the variable. For example, if a; be a variable, such expressions as 3^, 3 a" + 2, x1 + 4, e^ co", log x, are functions of x. A function thus defined is itself a variable whose values are determined when the value of a: is known. The variable x is called the independent variable, and the function is called the de- pendent variable.. A function in which the independent variable is affected by only a finite number of the algebraic operations, 40 GRAPHICAL REPRESENTATION 41 addition, subtraction, multiplication, division, involution, and evo- lution, is called an algebraic function. For example, the function q ^ _ /» ———- is obtained from the variable x by six algebraic opera- tions, viz. squaring x, multiplying by 3, subtracting 6, multiply- ing x by 2, adding 3, and, finally, dividing 3 a;2—6 by 2x+ 8. The functions aa-2 + bx + c and V25 — a-2 are likewise algebraic, while the function ('r\" v -r2 e^lim l+x\=l+x+——+..., n=» n) 11-2 which involves the consideration of an infinite number of alge- braic terms, is not algebraic, but belongs to a class known as tran- scendental functions. The function log a: is also transcendental. Algebraic functions are divided into several classes: rational functions, like -——— i which do not involve root extraction; j^ X ~f~ 0 rational integral functions, like 3 a;2+5 a-—6, which do not in- volve either division or root extraction; and irrational functions, like V25 — a-2, which involve root extraction. If it is not necessary to give a function explicitly, or if the character of the dependence is unknown, the symbol /(a-) is used to denote a function of x. Different functions are denoted by different letters or by the same letter with different sub- scripts. If f(x) == 3 a:2 + 6 x — 5, the function S^+Gy—Q would be represented byY(y), but the function 2 a:2+6 a;—5 would have to be represented by a different symbol, say, f,(x) or g (x). For the graphical representation of the relation between a variable and a function depending upon it, a second line, called {he function axis, or the axis of the dependent variable, is required. This second axis is drawn through the zero point of the first, at right angles to it, and the intersection of the two axes is taken as the zero point of the second as well as of the first. The two axes are together called coordinate axes. If the dependent vari- able, i.e. the function, be denoted by y, so that by definition y=/(a-), we can speak of the axis of the independent variable as the a:-axis, and of the function axis as the y-axis. Positive values of 42 MATHEMATICAL THEORY OF INVESTMENT
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the function are laid off above and negative values below the a;-axis, or, to speak more accurately, the positive part of the y-axis has the direction that would be taken by a line which is rotated about the origin in the direction opposite to that described by the hands of a clock, and through an angle of 90° measured from the positive part of the a;-axis. A line representing the function is laid off in a line parallel to the y-axis and from the extremity of the line representing the corresponding value of the inde- pendent variable. These two lines, representing, respectively, the independent variable and the function, will determine uniquely a point in the plane of the coordinate axes, and, conversely, any point in the plane of the axes deter- mines two lines, viz. the per- pendiculars to the two axes. A point in the plane of co- ordinate axes may be consid- ered without reference either to function or to variable. The ____ _____^_ _______ perpendicular from the point ° M to the a;-axis is called the ordi- nate of the point, and the dis- Fio. i tance from the origin to the foot of the ordinate is called the abscissa. The abscissa and the ordinate are together called the coordinates of the point. In the figure (Fig. 1) the line OM is the abscissa and the line MP the ordinate of the point -P. The terms abscissa, ordinate, and coordinate are used to denote either the lines determined by the point in question or the numbers indicating the lengths of the lines. In the latter case we write (x, y) to denote the coordi- nates of a point whose abscissa is a- and whose ordinate is y. To return to the graphical representation of a function of a variable, consider the function y=2a-+l. For any given value of x, say, x = 2, the value of y correspond- ing to it can be computed. "When x = 2, y == 5, and this pair of values determines a definite point in the coordinate plane. GRAPHICAL KEPRESEN'TATIOK
43
Similarly, for any value of x the corresponding value of y may be found, and the point determined by the pair of values may be located. The totality of all points so determined will lie on a line, either curved or straight, and this line is the graph of the function 2 a- +1, or, what is the same thing, the graph of the equation ,, ., i y=2x+l. The actual construction of the graph is best accomplished by forming a table of values of the independent variable, together
FIG. 2 with the corresponding values of the function, such as is shown herewith. The table may be extended at will in either direction, or a larger number of values of x and corresponding values of 'the function may be considered for any given interval. The points determined by each pair of values are then located, and a smooth curve is drawn through the points. In the present example the graph seems to be, as it really is, a straight line (see Fig. 2). In a similar fashion we may construct the graph of any function whose values corresponding to given values of the independent
44 MATHEMATICAL THEORY OF INVESTMENT variable may be computed. It may be that the function is such that to a given value of the independent variable more than one value of the function, or no value, will correspond. Such a function is r^-——; y=±V25-a:2; for if the number 3 be substituted for x, two values, +4 and — 4, are found for y. Clearly the substitution for x of any value lying between + 5 and — 5 will give two values for y. On the other hand, if a; =10, y=+V—75 or — V— 75, both of which are imaginary and cannot be represented by the method under consideration. The graph has, then, no real existence for values of x which are less than — 5 or greater than + 5. It will be found to be a circle with radius 5, having its center at the origin. It is not at all necessary that we should know the form of the function, provided we have a means of determining the values corresponding to the values of the independent variable. The function values may be determined, as they frequently are, by observation. Such examples would be the hourly thermometer readings for a day, the population of a city taken annually, the increase of a city's expenses by years. In fact, any set of observed data which varies with the time may be represented graphically by using the time as abscissa and the corresponding observed value as ordinate. ILLUSTRATIVE EXAMPLE. On a certain day, at a given station, the houriv thermometer readings, beginning with midnight, were observed to be 0°, -1°, - 3°, - 5°, - 7°, - 9°, - 8°, - 6°, - 4°, - 2°, 0°, 3°, 7°, 10°, 11°, 11°, 9°, 8°, 7°, 6°, 4°, 2°, 0°, - 2°, - 4°. What is the temperature curve for the day ? Solution. On the axis of abscissas lay off the values of the time from one hour to twenty-four, and at the extremities of the abscissas erect perpendicu- lars representing the observed temperatures. A smooth curve drawn through the extremities of these ordinates will be the curve required. Its form is shown in Fig. 3. It should be carefully noted that th