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. 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 1⁄2 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 x, 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 f(x) = (-x)2-4 = x2 - 4 = f(x). 1 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. = ± These ab ting y = 0 in (2), which gives x = scissas 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. 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. 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. The following theorems, whose proofs are left as exercises, follow from the facts that the points (−x, − y) and (x, − y) are symmetrical to the point (x, y) with respect to the origin and the x-axis respectively. Theorem 2A. The graph of f(x) is symmetrical with respect to the origin if Theorem 2B. The graph of an equation is symmetrical with respect to the origin if the equation obtained by replacing x by −x and y by −y, and simplifying, is identical with the given equation. Theorem 3B. The graph of an equation is symmetrical with respect to the x-axis if the equation obtained by replacing y by-y, and simplifying, is identical with the given equation. We shall use the phrase to discuss the table of values of a function to mean that the Symmetry, Values to be excluded, Intercepts, and Asymptotes (see next section) are to be determined before building the table of values. For the last three considerations, solve the equation for y in terms of x and for x in terms of y. But if the intercepts are desired independently, they may be found by setting either variable equal to zero and solving for the other. EXERCISES # 1. Does f(x) always equal either ± ƒ(x)? 2. Discuss the table of values (omitting asymptotes) and plot the graph of each of the functions and equations. 3. Discuss the table of values (omitting asymptotes) and plot the graph of (c) x2 + y2+ 4x 0. 4. If f(x) is any one of the functions whose graphs are given below, determine whether or not ƒ(− x) = ± f(x), find the value of ƒ(0), and the = # values of x for which f(x) is zero, and for which it is imaginary. 11. Functions becoming Infinite. Asymptotes. Continuing the discussion in the preceding section, in the following example we shall need the DEFINITION. It is said that a function becomes infinite as x approaches a if the numerical value of the function can be made larger than any positive number, however large, by giving x a value sufficiently near to a. |