TO THE TEACHER Many colleges now require for entrance examination a very elementary knowledge of the rudiments of the subject of Graphs. It is the opinion of many good teachers that an insight into Graphs is of considerable value to the pupil in finding the roots of equations, especially equations of the second degree and of degree higher than the second. All agree that the study of Graphs tends to stimulate the interest of the pupil in the work of finding the roots of equations. At the request of many teachers and superintendents this short chapter has been added to this treatise on elementary algebra. Five pages are devoted to giving the necessary definitions and explanations of the subject, and showing the pupil how to plot points. Four pages then treat of solving linear equations, six pages of solving quadratics, and one page of solving equations of degree higher than the second. A natural way is to study pages 409-417 after or with the chapter on Simultaneous Simple Equations, and to study pages 418-423 after or with the chapters on Quadratic Equations and Simultaneous Quadratics. 408 CHAPTER XXVII. GRAPHS. 436. Graphs. Diagrams, called graphs, are often used to show in a concise manner variations in temperature, in population, in prices, etc., etc. 437. Variables and Constants. A number that, under the conditions of the problem into which it enters, may take different values is called a variable. A number that, under the conditions of the problem into which it enters, has a fixed value is called a constant. NOTE. Variables are represented generally by the last letters of the alphabet, x, y, z, etc.; constants, by the Arabic numerals and by the first letters of the alphabet, a, b, c, etc. 438. Algebraic Functions. A function of a variable is an expression that changes in value when the variable changes in value. In general, any expression that involves a variable is a function of that variable. If x is involved only in a finite number of powers and roots, the expression is called an algebraic function of x. An algebraic function of x is rational and integral as regards x, if x is involved only in positive integral powers; that is, in powers and numerators, but not in roots or denominators. Thus, x2, Vx2 + x, 1 x3 + 4 are algebraic functions of x; but ax, Ve are not algebraic functions of x. Of ax2+br+c, the last three only are rational integral functions of x. For brevity a function of x is represented by f(x), F(x), (x), each of which is read function x. 409 439. As an easy example we may illustrate by a graph the changes in temperature for a day from 8 A.M. to 8 P.M. The official temperatures for Boston, July 17, 1905, were as follows: 8 A.M., 71°; 9 A.M., 72°; 10 A.M., 73°; 11 a.m., 77°; 12 M., 82°; 1 P.M., 85°; 2 P.M., 86°; 3 P.M., 88°; 4 P.M., 90°; 5 P.M., 89°; 6 P.M., 88°; 7 P.M., 86°; 8 P.M., 82°. Draw a horizontal line XX' and a line OY perpendicular to XX'. Using any convenient units of length, lay off on XX' equal distances to represent the hours and on OY equal distances to represent degrees of temperature from 70° to 90°. At each point of division on XX' draw a perpendicular of sufficient length to represent the temperature at that hour. Through the upper ends of these perpendiculars draw the line AB. This line, or graph, presents to the eye a complete view of the changes in temperature for the day. 440. Coördinates. Let XX' be a horizontal straight line, and let YY' be a straight line perpendicular to the line XX' at the point 0. Any point in the plane of the lines XX' and YY' is determined by its distance and direction from each of the perpendiculars XX' and YY'. The distance of a point from YY' is measured from O on the line XX' and is called the abscissa of the point. The distance of a point from XX' is measured from O on the line YY', and is called the ordinate of the point. Thus, the abscissa of P1 is OB1, the ordinate of P1 is OA1; The abscissa and the ordinate of a point are called the coördinates of the point. The lines XX' and YY' are called the axes of coördinates, or the axes of reference; the line XX' is called the axis of abscissas, or the axis of x; and the line YY' is called the axis of ordinates, or the axis of y. The point is called the origin. In general, an abscissa is represented by x, and an ordinate by y. The coördinates of a point whose abscissa is x and ordinate У are written (x, y). In this notation the abscissa is always written first and the ordinate second. Thus, the point (4, 7) is the point whose abscissa is 4 and ordinate 7. Abscissas measured to the right of YY' are called positive, to the left of YY' are called negative; ordinates measured above XX' are called positive, below XX' are called negative. Thus, in the figure on page 411 the point P1 is (8, 5), the point P2 is (6, 3), the point P3 is (-4, — 3), and the point P4 is (5, — 4). 441. Quadrants. The axes of coördinates divide the plane of the axes into four parts called quadrants. The quadrant XOY is called Quadrant I, the quadrant X'OY is called Quadrant II, the quadrant X'OY' is called Quadrant III, and the quadrant XOY' is called Quadrant IV. Every point in Quadrant I has a positive abscissa and a positive ordinate; every point in Quadrant II has a negative abscissa and a positive ordinate; every point in Quadrant III has a negative abscissa and a negative ordinate; every point in Quadrant IV has a positive abscissa and a negative ordinate. Hence, the signs of the coördinates of a point show at a glance in what quadrant the point is situated. 442. Plotting Points. It is evident that if the location of a point is known, the coördinates of that point referred to given axes may be found easily by measurement; and if the coördinates of a point are given, the point may be readily constructed, or plotted. Thus, a convenient length is taken as the unit, and the point P is found by measurement to lie 2 units to the right of YY' and 4 units below XX', and is, therefore, the point (2, − 4). Again, to plot the point (5, 2), a distance of 5 units is laid off on XX' to the left from O to K1, and a distance of 2 units on YY' upwards from 0 to K2. The intersection of the perpendiculars erected at K1 and K2 determines the point K, which is the required point (5, 2). |