Using this association of points and numbers, if P1 and P2 are two points on the line, and x1 and x2 are the numbers associated with them, we have from (2) This difference of the values of the x's is denoted by Ax, so that Ax =P1 P2 = X2 - X1 (4) Notice that Ax is a single symbol (never the product of two numbers ▲ and x), and that its value is obtained by subtracting the value of x corresponding to the first point from that corresponding to the second. EXERCISES 1. Illustrate (3) by numerical examples for the six possible relative positions of O, P1, and P2. Find Ax for each case. 1 2. Show that Ax is positive or negative according as P1 lies to the left or right of P2, using the definition that a<b if b – a is positive. 3. If OP = x, show that x increases or decreases according as P moves to the right or left. 8. Rectangular Coördinates. Let X'X and Y'Y be two are called rectangular coördinates of P, x the abscissa, and y the ordinate. In the figure, the ordinate y is often thought of as MP, which equals ON in both magnitude and direction. The directed lines X'X and Y'Y are called the axes of coördinates and their intersection O the origin. The abscissa of a point is positive or negative according as the point lies to the right or left of the y-axis. The ordinate is positive or negative according as the point lies above or below the x-axis. In the figure, the abscissa x is positive, while the ordinate y is negative. As seen above, any point P determines a pair of real numbers, its coördinates. Conversely, given any pair of real numbers, x and y, a point may be plotted, that is, constructed, whose coördinates are x and y. For on X'X lay off OM = x. At M erect a line perpendicular to X'X and on it lay off MP = y. Then P is the required point. The symbol (x, y) is used to mean the point whose coördinates are x and y. If P is this point, it is indicated by the symbol P(x, y). Coördinate axes divide the plane into four parts called quadrants. These are numbered as in the figure, which also indicates the signs of the coördinates of a point in every quadrant. We are frequently concerned with points which are symmetric with respect to the origin, the axes, or a line bisecting the first and third quadrants. Points having these symmetric relations are determined in accordance with the DEFINITIONS. (1) Two points are symmetric with respect to a YA II (−, +) I (+,+) X X III (--) IV (+;-) FIG. 13. third point if the line joining the two points is bisected at the third point. (2) Two points are symmetric with respect to a line, if the line is the perpendicular bisector of the line joining the two points. EXERCISES 1. Plot the points whose coördinates are given below, and determine the nature of the symmetry for the pairs of points in each group. (a) (2, 1), (2, 1); (3, 0), (3, 0); (c) (2, 1), (d) (2, 1), (-2, -3), (2, 3). 2. By means of the corresponding parts of Exercise 1, what can be said of the positions of the following pairs of points? (a) (x, y) and (− x, y). (c) (x, y) and (— x, − y). (b) (x, y) and (x, − y). 3. One end of a line bisected by the origin is the point (− 5, 2). What are the coördinates of the other end? 4. What are the coördinates of the point symmetrical to the point (-3, 4) with respect to the y-axis? the x-axis? the origin? the line bisecting the first and third quadrants? 5. Find the coördinates of the vertex or vertices not given in the regular polygons located as follows: (a) One vertex of an equilateral triangle is the point (1, 0) and the altitude through this vertex, which is √3 units long, extends through the origin. (b) An equilateral triangle has vertices with coördinates (0, 0) and (1, 0). (c) A square with opposite vertices having coördinates (1, 0) and (-1, 0). (d) A hexagon two of whose opposite vertices have coördinates (1, 0) and (-1, 0). (e) An octagon with two opposite vertices having coördinates (1, 0) and (-1, 0). 6. The coördinates of three vertices of a rhombus are (-1, 0), (0, √3), (1, 0). What are the coördinates of the fourth vertex? (Three solutions.) What are the coördinates of the intersection of the diagonals? 9. Graph of a Function. Values of x and the corresponding values of a function may be exhibited in tabular form. Table 1 gives the population of the United States in millions for the successive decades from 1830 to 1910. Table 1. X 1830, 1840, 1850, 1860, 1870, 1880, 1890, 1900, 1910, Y 12.8, 17, 23.1, 31.4, 38.5, 50.1, 62.6, 75.9, 93.9, Table 2 gives pairs of values of x and the function x + 5· Any such table of values may be strikingly exhibited to the eye by plotting the points whose coördinates are the pairs of numbers in the table, and then drawing a smooth curve through the points so obtained. Proceeding thus, we obtain from Tables 1 and 2 the curves in Figures 14 and 15, which are called the graphs of the functions. DEFINITION. The graph of a function is a curve such that (1) Any point whose coördinates are corresponding values of x and the function is on the curve, and (2) Conversely, the coördinates of any point on the curve -90 -80· -70 -60· -50 -40 -30 -201 -10 1830 1840 1850 1860 1870 1850 1890 1900 1910 X FIG. 14. are a pair of corresponding values of x and the function. Hence the important fact: which have no graphs, and the graphs of others are merely one or more isolated points, but we shall not encounter them in this course. The graph of a function may be constructed by the following process: Construct a table of values of x and the function of x. Plot the points whose coördinates are the pairs of numbers in this table. Draw a smooth curve through these points. In constructing a graph, notice that values of a giving imaginary values of the function are discarded, and that the number of points plotted must be large enough to indicate without doubt the form of the curve. Whenever it is not clear just how the curve is to be drawn, enlarge the table of values, either by giving more integral values of x, or by assuming for x intermediate values such as 2.5, 2.8, etc., as may best serve the purpose. The necessity for this last remark is shown by the fact that three points, situated as in Fig. 16(a), can be connected as in Figs. 16(b), (c), (d) for different types of function, and also in other ways (see Exercise 4 below). DEFINITION. The graph of an equation in two variables is the curve such that: (1) Every point whose coördinates satisfy the equation is on the curve, and (2) Conversely, the coördinates of any point on the curve satisfy the equation. To plot the graph of an equation, Solve the equation for one of the variables in terms of the other, thus obtaining one as a function of the other. Then proceed as indicated in the rule for the graph of a function. In many of the applications of the methods of coördinates, the coördinates refer to quantities of different kinds such as time, distance, work, cost, etc., and the graph represents a relation between two of these quantities. EXERCISES 1. Construct the graphs of each of the following pairs of functions on the same axes. State a relation that each pair of graphs bear to one another. (a) 3x, 3x+2. (b) 2x, −2x+2. (c) 1x, 1x −3. |