things besides the time, such as the season of the year, the lati- tude and the altitude of the station, and so on. Indeed, in prob- lems of the second kind the relation of ordinate to abscissa can scarcely be called a functional relation. It is, rather, a mere corre- lation of values. The curve is not for that reason without value. The method is largely used in the theory of statistics, which is applied to the investigation of a variety of important problems. r
x
FIG. 3 18. Some important graphs. 1.The graph of tJie linear integral function. A rational integral function (see § 17) in which the variable occurs in the first degree and no higher is called a linear integral/unction. It is necessarily of the form ax+b, where a and b are constants, so that v= ax+b - C1') is the equation between the function and the independent variable. The reason for the use of the term linear is given by the following THEOREM : The graph of a linear integral function, or, what amounts to the same thing, the graph of an equation of the first degree between two variables, is a straight line.
46
MATHEMATICAL THEORY OF INVESTMENT
Let P be any point on the graph, B the intersection of the graph with the y-axis, and A the intersection of the graph with the a:-axis, as in Fig. 4. Draw the ordinate MP of the point -P, and draw a line BR from B parallel to the a--axis and meeting
FIG. 4 the ordinate MP in -E. The abscissa of the point B is 0. Sub- stituting this -value of a; in the equation y = ax + b, we find for the value of the ordinate of the point B y=^ i.e. the length of the line OB is b. Moreover, PR = MP - MR == MP - OB =y-b. But BK = OM= x. Hence the ratio of the two lines RP and BE, is
But, from equation (1),
RP _y-b JSR~ x
y-b
x
=ffl,
(2) GRAPHICAL REPRESENTATION 47 as is easily seen when b is transposed to the left member and the whole equation divided through by a-. Consequently, Ij- w This relation may be translated into words by saying that the ratio of the distances of the point P from two fixed lines, viz. BR and BY, is constant. From geometry we know th; A the locus of a point, the ratio of whose distances from two fixed lines is con- stant, is a straight line. Therefore the graph of the equation y = ax + b, or of the linear function ax + b, is a straight line, as the theorem asserts. Referring to Fig. 4, we see that the two lines BR and KP, which occur in the equation (3), may be likened to the tread and the riser of one of the steps of a stairway. It is obvious that the steepness of the stairway depends upon the ratio of the length of the riser to the length of the tread. If the riser E.P is long in comparison with the tread BR, the stairway will be steep, while if it is short in comparison with BR, the slope will be gentle. In other words, the value of a determines the steepness of the line. For this reason a is called the slope of the line. The term slope is used in exactly the same way as a carpenter uses the term pitch when speaking of the steepness of a roof. When the graph is not a straight line, the slope at any point is defined as the slope of the tangent at that point. The slope of a curved line is not less important than that of a straight line. It is, however, beyond the scope of this book to take up the discussion of the slope of a curve, which is a matter belonging to the differential calculus. 2. The graph of the quadratic integral function. A quadratic integral function is an algebraic function which is rational, inte- gral, and of the second degree. Its form is OKB2 + bx + C, and it has the same graph as the equation y=axi+hx+c. (4) 48
MATHEMATICAL THEORY OF INVESTMENT
The graph is a well-known curve called a parabola. The graph of the quadratic function a-2— 5a;+ 6 y is shown in Fig. 5. -———\—————-f—- 3. The logarithmic -————————'—- function. The graph ____\_____A___ of the function loga; ____Ji____/___ or the equation ___^i___jf___ y=\ogx (5) ____J L_- is nothing more or -————-^-—/__ less than a graphical _______\_/ |____^. , ,. ,. o ^7 x representation ot a __________ portion of a table of _____ _ logarithms forming a given system. The ' g^ of Fie. 5 y = ^Si^ is given in Fig. 6. No computation is required if a table of common logarithms is at hand, and tlie construction of the curve presents no difficulty.
FIG. 5
Fio.6
All logarithmic curves whose equations have the form (5) pass through the point (1, 0). Why is this statement true ? GRAPHICAL REPRESENTATION
49
4. The exponential function. An exponential function is one in which the variable occurs as an exponent. The simplest form of such a function is a", though the name, exponential function, is usually reserved for the special function e", where e denotes the Napierian base 2.71828 .. The table of values is easily constructed for integral values of the exponent when the base is an integer. For a =2 the table
x
FIG. 7 of values is as shown in the accompanying table, and the curve, which is the graph of the equation y=2% is that shown in Fig. 7. All exponential curves whose equations are of the type y ^ ^ ^) pass through the point (0,1). Why ? There are many phenomena which follow a law whose mathe- matical statement is equation (6). Such phenomena are said to follow tile compound interest law, a law which is of very great importance in many branches of pure and of applied mathematics as well.
50 MATHEMATICAL THEORY OF INVESTMENT EXAMPLES 1. Construct the graphs of the functions x, 2x, 3x, and 4 a: with the same coordinate axes. 2. Construct the graphs of the functions 2 x — 1, 2 a;, 2 x +1, and 2x + 2 with the same coordinate axes. 3. Assuming that the graphs in Example 2 are straight lines, prove that they are parallel. 4. Construct the graphs of the equations y=^, y=^-2)2, Sy^a-Sa'+e. 5. Construct the graph of x2 + y2 = 25. 6. Construct the graph of the function y = x3. 7. Beginning with the year 1876, the high-school attendance in this country was 22,982; in 1880 it was 26,609 ; and at the end of the succeed- ing ten-year periods up to 1910 it was as follows: in 1890, 202,963; in 1900, 519,251; and in 1910, 915,061. Construct the graph showing the high-school attendance for these years. What was the approximate attend- ance in 1895 ? What is it likely to be in 1920 ? 8. Construct the exponential curve whose equation is y = S". 9. From the curve in Example 8 find the logarithm of 10 to the base 3. 10. Construct the curve of the function A when A is given by the equation A = (1.05)". 11. Construct the graph of the function 100 x .05 x (, and show how it may be used as a table for simple interest at 6%. 12. Construct the graph of the function