8. Specific gravity is the ratio of the density of a body to that of another body taken as a standard. Water which is taken as the standard has a density of unity in the metric system. Hence the mass of a substance is equal to the specific gravity times the volume. How much water must be added to 25 cubic centimeters of concentrated hydrochloric acid, specific gravity 1.20, to reduce the specific gravity to 1.12? 9. How much water must be added to 30 cubic centimeters of ammonia, specific gravity 0.90, to raise the specific gravity to 0.96? 10. It is desired to reduce the specific gravities of quantities of sulphuric acid, specific gravity 1.84, and of nitric acid, specific gravity 1.42. How much water must be added to 80 cubic centimeters of each to reduce the specific gravities to 1.18 and 1.20 respectively? 11. What mass of hydrogen (molecular weight 2) will be displaced by 10 grams of zinc (molecular weight 65) acted upon by hydrochloric acid if the amounts exchanged are proportional to the molecular weights? 12. Find the money value of pure gold in a $20 gold piece, if one ounce of gold is worth $20.66 and the coin weighs 516 grains and is pure. 13. A tank can be filled by one pipe in 3 hours, and emptied by a second in 2 hours, and by a third in 4 hours. How long will it take to empty the tank when it is full if all the pipes are opened? 14. The report of a gun was heard in 3 seconds at a place 3189 feet distant, toward which the wind was blowing; and in 2 seconds at a place 2074 feet distant, from which the wind was blowing. Find the velocity of sound and of the wind. 15. A railway passenger observed that a train moving in the opposite direction passes him in 2 seconds, but when moving in the same direction it passed him in 30 seconds. Compare the rates of the trains. 16. Two cities are 39 miles apart. If A leaves one city two hours earlier than B leaves the other, they meet two and a half hours after B starts. Had B started at the same time as A, they would have met three hours after they started. How many miles an hour does each walk? 17. The accuracy with which one can weigh an object on a balance depends on the number of spaces on the scale which the pointer moves W 5, 25, when a small weight is added. The table gives the deflection, D, of the pointer with a weight W in the pan when one milligram is added to W. Determine D, approximately, as a function of W. CHAPTER III ALGEBRAIC FUNCTIONS 28. Introduction. In this chapter we shall study the properties (Sections 10 to 13) of certain types of algebraic functions (Section 14). Among these types are the quadratic function, ax2 + bx + c, which occurs in the solution of quadratic equations, and the function ax". The latter is an integral rational function if n is a positive integer, a rational fractional function if n is a negative integer, and an irrational function if n is a fraction. Other types are the linear fractional function, (ax + b)/(cx + d), and polynomials, which are studied in order to obtain a method for solving equations of higher degree than the second, an extension of the solution of linear and quadratic equations. These functions find frequent application in many fields. For example, the quadratic function appears in the theory of falling bodies, and among the laws which can be represented by a function of the type ax" are Newton's law of gravitation, and Boyle's law connecting the pressure and volume of a gas. In the study of quadratic functions we shall proceed from special cases to the general case by methods which are important in other connections. 29. Graph of x2. Since (-x)2 = x2, the graph is symmetrical with respect to the y-axis (Theorem 1A, page 23). Hence the y-axis is called an axis of symmetry. only positive values of x. x y -9 -8 7 FIG. 47. 0, 1, 2, 3, x2 0, 1, 4, 9, The table of values need include It is readily seen that the intercepts are x = 0 and y = 0. Hence the curve passes through the origin, but does not cut the axes elsewhere. Since x2 is positive if x is real, no part of the curve lies below the x-axis. As x increases, so also does x2, and hence the curve broadens out as it rises. If x > 1, x2 increases more rapidly than x, so that, to the right of x = 1, the curve rises more rapidly than it broadens out. The origin is a minimum point. 7 -6 30. Graphs of ax2 and af(x). Consider the graph of the function 2x2. The ordinate of any point on the graph is twice that of the point with the same abscissa on the graph of x2. Hence the graph of 2x2 may be plotted by doubling the ordinates of a number of points on the graph of 22, and drawing a smooth curve through the points so obtained. FIG. 48. a In like manner, the graph of 3x2 may be obtained by trebling the ordinates of points on the graph of x2, of x2 by bisecting them, etc. The ordinates of points on the graph of - 2x2 are numerically twice those of points on the graph of x2, but as they are negative, the points on the graph lie below the x-axis. The graphs of 2x2 and - 2x2 are symmetrical to each other with respect to the x-axis. From the method of constructing these graphs we see that The y-axis is an axis of symmetry of the graph of ax2. The curve runs up or down, and the origin is a minimum or maximum point, according as a is positive or negative. If a is positive, the graph of ax2 rises more or less rapidly than that of x2, for which a = 1, according as a is greater or less than unity. The larger the value of a, the more rapidly the curve rises, and the smaller the value of a is, the less rapidly it rises. Any one of these curves is called a parabola. The axis of symmetry of a parabola is called its axis. The reasoning employed above may be used to prove the Theorem. The graph of af(x) may be obtained by multiplying by a the ordinates of points on the graph of f(x). Corresponding points on the two graphs lie on the same or opposite sides of the x-axis according as a is positive or negative. If a = 1 the graph is symmetrical to that of f(x) with respect to the x-axis. This theorem is the second one we have considered belonging to the set of relations between pairs of functions and their graphs (see the last paragraph but one on page 43). 31. Translation of the Coördinate Axes. Consider a system of coördinate axes with the origin O, and a second system, parallel to the first, with origin O'. Replacing the first system by the second is called translating the axes. By the equations for translating the axes we mean equations which express the coördinates of a point referred to the first, or old, axes in terms of the coördinates of the same point referred to the second, or new, axes. Let the old and new coördinates of any point P be respectively (x, y) and (x', y'), and let the old coördinates of the O new origin, O', be (h, k). From the figure we readily obtain x' FIG. 49. Theorem 1. The equations for translating the axes are = x' + h, 第ˊ} y = y' + k, where (h, k) are the coördinates of the new origin. (1) Suppose that the graph of an equation in x and y has been plotted. To find the equation of this graph referred to new axes parallel to the old axes we substitute in the given equation the values of x and y given by (1). The following examples illustrate the utility of the theorem. EXAMPLE 1. Given the equation y = x2 6x + 13, (2) translate the axes so that the new origin is the point (3, 4) and find the equation in the new coördinates which has the same graph as (2). Plot the graph, and state its most noteworthy properties. FIG. 50. Removing the parentheses, or y' + 4 = x22 + 6x′ + 9 − 6x' 18+ 13, y' = x22. (5) The graph of (5), plotted on the new axes, is the same as the graph of (2), referred to the old axes. But the graph of (5) is a parabola which is easily plotted on the new axes. It is shown in the figure. This curve is also the graph of (2) when plotted on the old axes. From it we see that the axis of symmetry of the parabola, the y'-axis, is the line x 3, and that the minimum point is the new origin (3, 4). In this example it turned out that the equation obtained by translating the axes was much simpler than the given equation. The question arises: If an equation is given, how can we determine how to move the axes so as to obtain a simpler equation with the same graph? The method of doing this is illustrated in by translating the axes, and construct the graph. (6) First solution. Substituting in (6) the values of x and y given by (1), This equation will be very simple if the coefficient of x' and the constant term are zero, that is, if |
