axes. 2. Construct the graphs of the following pairs of functions on the same Can the graph of one member of a set be moved so as to coincide with the graph of the other member? If so, how can this be done? (b) x2 + 4x + 4, x2 + 4x. (c) — x2, − x2 + 4. (a) 2x2, 2x2 + 3. 3. Prove the Theorem: The graph of f(x) + k may be obtained by moving the graph of f(x) a distance k in the direction of the y-axis. 4. Plot the graphs of each of the functions in the following sets. To insure the proper connection of points obtained from integral values of x use intermediate values of x. (a) x3 − x2, 2x3 (b) – x3 + x2, 3x2 + x, 2x3 x2 5. Plot the graphs of the following equations: X. 2. 6. Distribution of the heights of 12-year-old boys. state one of its characteristics. Plot the graph and Heights 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62. in inches. Number of boys. } 1, 6, 18, 27, 45, 89, 115, 148, 123, 76, 34, 13, NOTE. In constructing the graphs of some functions it may be desirable to choose different units of length on the x- and y-axes. 7. The following tables give the monthly receipts of eggs in the Chicago market in 1910, the price per dozen, and the storage of eggs by months in percentages of the total annual storage by a Chicago firm. Month. Eggs in thousand cases. Storage of eggs in percentages. J. F. M. A. M. J. J. A. S. O. N. D. 72, 140, 160, 760, 500, 400, 300, 240, 180, 120, 80, 48. 1.5, .8, 9.5, 42, 19, 22, 1, 0, 1, .3, .3, 2.6. dozence per.} 27, 27, 23, 21, 20, 18, 17, 20, 23, 26, 30, 33. Let time be the independent variable in each instance, and plot the three graphs on the same set of axes. Let a convenient length from the origin on the y-axis represent the three values, 800 thousand cases, 100%, and 40 cents a dozen, then mark off the subdivisions 50 thousand cases, 5%, and 5 cents a dozen, and multiples of these subdivisions. State a relation that exists between each two of the three graphs. What inferences can be drawn from the graphs with respect to the effect of storing eggs on the price? 8. The following are the monthly statistics for butter received, stored, and the price in the Chicago market in 1910. F. M. A. M. J. J. A. S. O. N. D. ceived in 40, 52, 60, 72, 108, 168, 148, 104, 112, 96, 72, 56. thousands. Storage in Storage in 6, 3.6, 2.8, 3.4, 14, 43, 10, 2, 4.2, 2.5, 4.5, 4. percentage.} Price per lb. in cents. 26, 26, 28, 27, 29, } 34, 28, 31, 31, 26, 26, 26, Plot the three graphs on the same set of axes, as in problem 7. State a relation that exists between each two of the three functions. What inferences can be drawn from the graphs regarding the effect of storage of butter on the price? NOTE. When a law is stated in such general terms that numerical data representing a concrete situation cannot be derived, a graph which will picture the general situation can be obtained by constructing a table of values with purely arbitrary sets of numbers which conform to the law. Such a graph will indicate the mode of change of the function with respect to the independent variable without representing a concrete situation. 9. The Weber-Fechner law of psychology states that as the intensity of an external stimulus increases in geometric progression, the corresponding sensation increases in arithmetic progression. Construct the graph. Let x represent the external stimuli, and y the corresponding sensations. Let the geometric progression 1, 2, 4, 8, . . be the values assumed for x, and the arithmetic progression 1, 2, 3, 4, be the values assumed for y. Plot the points whose coördinates are (1, 1), (2, 2), (4, 3), etc., and draw the graph. At what value of x should the graph begin? Can there be an external stimulus without a corresponding sensation? The law is said to hold for the senses of touch, hearing and seeing, but not for taste and smell. 10. Water pressure dies away uniformly because of the resistance of the conduits. Construct a graph to show the change in water pressure for points at different distances from a reservoir situated on a hill. 11. Malthus' law states that population increases in a geometric progression with reference to time, while subsistence increases in arithmetic progression. Plot and discuss the graphs with reference to the influence these two relations have on one another. The law of diminishing returns states that after a certain point, doubling the cultivation in agriculture will not double the returns. How will this affect the preceding graphs? NOTE. In the following problems the functional relation changes in character at two points, and the graph of the function consists of several distinct parts. 2 12. The amount of heat required to raise through one degree the temperature of one gram of ice is a calorie, of one gram of water is one calorie, of one gram of steam is a calorie, approximately. 80 calories of heat are absorbed without any rise of temperature when the ice is melting, and 537 calories without rise of temperature at the boiling point when the water is vaporizing. If the quantity of heat absorbed is regarded as a function of the temperature xæ, construct a graph representing roughly the change from ice at 10° below freezing to steam at 5° above boiling. (Centigrade scale.) 13. In the case of mercury the amount required to raise one gram 1° in any of the three forms is approximately .033 calorie, the fusing point is -38°, the heat absorbed in fusing 8.8 calories, the boiling point 675°, and the heat absorbed in vaporization 67.7. Construct a graph for mercury analogous to that for water. 14. A man walks away from his home at the rate of 4 miles an hour for three hours, and then returns at the rate of 2 miles an hour. Construct a graph showing his distance from home at any time. 15. A man rides away from a town at the rate of 6 miles an hour for 2 hours. He then stops for one hour, and walks back at the rate of 3 miles an hour. Construct a graph showing his distance from town at any time. 16. Construct on the same axes the graphs of the functions of x which give the perimeter and area of a square whose side is x. Determine from the graphs for what values of x the perimeter is (1) less than the area, (2) equal to the area, (3) greater than the area. 17. Construct on the same axes the graphs of the functions which express the circumference and area of a circle in terms of the radius. Determine from the graph for what values of r the circumference is (1) less than the area, (2) equal to the area, (3) greater than the area. 10. Discussion of the Table of Values. The considerations in this section and the section following enable us, in many cases, to abridge the labor of building a table of values, to overcome special difficulties, and to discover properties of the graph. EXAMPLE 1. Construct a table of values and the graph of f(x) = x2 - 4. Symmetry. We shall first see that the table of values need be computed only for positive values of x. Substituting -x for x, we have ƒ(−x) = 1 (−x)2—4 = 1⁄2 x2 − 4 = ƒ(x). Hence the function has the same value for any two values of x which are equal numerically, but differ in sign, and therefore if (x, y) is a point on the graph, so also is (--x, y). These points are symmetrical with respect to the y-axis, and hence the graph is also, in accordance with the DEFINITION. A curve is said to be symmetrical with respect to a line (or point) if its points by pairs are symmetrical with respect to that line (or point). The line (or point) is called an axis (or center) of symmetry. Then if the part of the curve to the right of the y-axis is plotted, the part on the left may be plotted by means of the symmetry, and hence only positive values of x are needed in the table. Now set Values Excluded. We have agreed to neglect imaginary values of x and y. If we substitute any real value of x in (1), we obtain a real value for y, and hence no values of x need be excluded. But from (2), we see that all values of y < 4, for example y = 5, make x imaginary. Hence if a table of values be constructed from (2) by assuming values of y, all values of y less than 4 must be excluded. Graphically, since no values of x are to be excluded, the curve runs off indefinitely to the right and left. Since no positive values of y are excluded the graph runs up indefinitely, but as values less than - 4 are excluded, no part of the curve lies more than 4 units below the x-axis. Intercepts. The coördinates of the points in which a graph cuts the axes are usually of special significance, and they should be included in the table of values. For points on the x-axis, y = 0, and hence the abscissas of the points where the graph cuts the x-axis are obtained by set = ± 2.8. ting y = 0 in (2), which gives x These abscissas are called the intercepts on the x-axis in accordance with the DEFINITION. The intercepts of a curve on the x-axis are the abscissas of the points where the graph cuts the x-axis, and the intercepts on the y-axis are the ordinates of the points of intersection with the y-axis. Since x = 0 for all points on the y-axis, the intercepts on the y-axis are found by setting x = 0 in (1), which gives y = — 4. DEFINITION. A zero of a function is a value of x for which the function is equal to zero. Hence the zeros of f(x) are identical with the roots of the equation f(x) = 0. All the zeros of a function which are real numbers are represented by the intercepts of the graph on the x-axis. F F Y We now build the accompanying table and plot the points A, B, C, D, E, F, from it. Then construct B' symmetrical to B with respect to the y-axis. This is done readily on crosssection paper by counting the squares from B to the y-axis and proceeding an equal number of squares beyond. Similarly, construct C', D', E', F' symmetrical to C, D, E, F respectively. Then draw the graph. The reasoning employed in 0-4 E 0 C 3 0.5 B 4 4 √8 FIG. 17. the illustration of symmetry is general, and may be used with any function or equation. Hence the theorems: Theorem 1A. The graph of f(x) is symmetrical with respect to the y-axis if ƒ(− x) = f(x). Theorem 1B. The graph of an equation is symmetrical with respect to the y-axis if the equation obtained by replacing x by x, and simplifying, is identical with the given equation. |