The points corresponding to these pairs of values of x and y are the intersections of the two graphs (Fig. 3), namely, P, Q, R, S. NOTE 1.-In case the graph of an equation of the second degree in two variables is a closed curve, the graph is called an ellipse. For example, 12x2 + 13y2 = 248 is the equation of an ellipse. NOTE 2-In case the equation has the form x2 + y2 = 16 42, the curve is a circle whose radius is 4. and 458. Graphs of miscellaneous quadratic forms. (1) x2+ y2-2xy -4x-8y-200 The first equation may be written xy=- 2. (1) y2 — 2 (x + 4) y + (x2 - 4 x 20) = 0. The student should note that, in equation (2), when x is-, y is +; and that, as x approaches 0 through negative values, y is + and approaches +. Similarly, when is, y is; and as a approaches 0 through positive values, y approaches -. as a passes through 0 from positive to negative values, y changes sign from - to +∞. Thus 1 To plot the points corresponding to pairs of values of x and y (two values of y for each value of x) in the first table, one point is located for each pair of values, i. e., two points for every value of x. The graph corresponding to the first table is the parabola in Fig. 4. The points corresponding to the first values of y are found in that part of the graph represented by BPA, and to the second values of y are found in that part of the graph represented by ACD. The points corresponding to x = 0, y = ±21—3, are not on the parabola. 2. The graph of xy=-2 is shown in Fig. 4; it has two branches extending to infinity, one lying in the angle YOX' and the other in the angle XOY'. The curve is therefore an hyperbola. On eliminating y between equations (1) and (2) we get x4x3-16 x2+16x+4= 0. The values of x in this equation are the abscissae of the points of intersection of the curves (1) and (2), namely Q, P, R, S; but the equation can not be factored and we have not yet had a method for solving an equation of the fourth degree; however, a careful plotting of curves (1) and (2) shows approximately what the values of x and y are which satisfy the given equations (Fig. 4), namely, Determine the graphs of the following systems of simultaneous equations in x and y in Exercise LXXVII, and locate the points represented by their solutions: Examples 5-12, 27-32, 37-40. CHAPTER XIII GRAPHS OF QUADRATIC EXPRESSIONS AND PROBLEMS IN MAXIMA AND MINIMA WHICH CAN BE SOLVED BY EQUATIONS OF THE SECOND DEGREE 459. If a variable quantity y, which, having increased continually for a given time, then decreases continually, passes through a value greater than its neighboring values, i. e., those which immediately precede and those which immediately follow, it is said that y passes through a maximum. On the contrary, if a variable quantity y, which, having decreased continually for a given time, then increases continually, passes through a value less than those which immediately precede and those which immediately follow, it is said that y passes through a minimum. Consider, for example, the sections of a series of ridges and valleys made by a vertical plane. Suppose this section to be represented by the curve MN, and that the heights of the different points of this curve above the horizontal plane PQ are measured. The summit, A, of a ridge will be a maximum and the bottom, B, of a valley will be a minimum. If one travels throughout the length of the curve, he will ascend till he reaches the point A, then he will descend till he arrives at B; the height A'A of the point A is greater than that of the neighboring points which immediately precede or follow; the point A is therefore a maximum. On proceeding from The A he descends to the point B, then ascends till is reached. height B'B of the point B, is less than that of its neighboring points either to the left or to the right; B'B is therefore a minimum. Similarly CC will be a second maximum and D'D a second minimum; and so on. Some simple problems will be investigated which can be solved by the equation of the second degree or by simple polynomials. 460. PROBLEM I.-Consider the variation of the product of two quantities whose sum is a constant a. Let x be one of the quantities and a x the other, of which the sum a x+x= a; and let y be their product whose variation will Here, then, be studied. (1) = y = x(a — x). It is seen that y 0, for x = 0 or for x = a. Hence, as x increases continuously from 0 to a, y increases continuously to a certain value and then decreases to zero. All of these values of y are finite, since none of the values of c is greater than a. It is seen, therefore, that y must, by definition, pass through a maximum value for some value of a greater than zero and less than a. To decide what value or values of <a and >0 will make y a maximum, write (1) in the form, On inspecting this formula, it is seen that y will have the greatest value for 0 < x <a, when the least quantity is subtracted from a2 i. e., 4 a a y = = a 2 2 2 4 = 0. a This will happen when x = =2; therefore, Hence, the product of two factors whose sum is a, will be a maximum when these two factors are equal to each other. It is easy to follow the variations of the product y as x varies from 0 to x = a. When x increases continuously from x = 0 to x = the term to be subtracted becomes smaller and smaller and 2 finally becomes 0 for x = =; the product y increases continuously from zero to the maximum value a2 When a becomes greater than a a 4' and increases until it becomes equal to a, the term to be subtracted, (-) or (-), increases and more, and y a decreases from to zero. a2 more |