NOTE. To find negative roots by Horner's method, replace x in the equation by x. The graph of ƒ(− x) is symmetrical to that of f(x) with respect to the y-axis, and the roots of the equation ƒ(− x) = 0 will be equal numerically to those of f(x) = 0, but have opposite signs. Hence the negative roots of an equation f(x) = 0 may be found by finding the positive roots of f(x) = 0, and changing their signs.

8. x3 + 2x + 23 = 0.

9. x3 + x2 + 7 = 0.

11. x4 3x3 4x2 + 12x - 10 = 0. 12. x4 2x3 — 7 = 0.

10. x3 x2-6x + 1 = 0.

NOTE. If an equation has both rational and irrational roots, it is advisable to find the rational roots first. Suppose they are a, ẞ, Y, etc. Then divide the equation by x a, the resulting equation by x 6, the new equation by x-y, etc., thus obtaining a simpler equation whose irrational roots are the same as those of the given equation, and then solve this simpler equation by Horner's method.

18. 6x5+17x4 + 4x3 39x2 - 297x + 210 = 0.

19. Find the fifth root of 279.

20. A cast iron rectangular girder (breadth =

depth) rests upon

supports 12 feet apart and carries a weight of 2000 pounds at the center. In order that the intensity of the stress may nowhere exceed 4,000 pounds per square inch, it is determined that the depth d of the girder in inches must satisfy the equation 80ď3 81d2 - 17,280 = 0. Find d and the crosssectional area.

21. The depth of flotation of a buoy in the form of a sphere is given by the equation x3 3rx2 + 4r3s = 0, where r is the radius and s is the specific gravity of the material. What is the depth for such a buoy whose radius is 1 foot and specific gravity is 0.786?

22. The cross section of the retaining wall of a reservoir is designed as indicated in Fig. 73. The allowable height x of the upper portion is given by the equation x3 + 32x2 69x-88 0, where x is expressed in terms of a unit of 10 feet. Find the allowable height to three significant figures.

}

23. The allowable height of the lower portion of the wall in Exercise 22 is given by the equation y1 - 2.7y3 + 18.4y2 - 123y – 1.46 = 0, where y is in terms of a unit of 10 feet. Find the height of the lower portion to three significant figures.

E

D C

100

24. In Exercise 22 the slope of the water front BC to the vertical is, of DE is, of AE is 70%. The width of the top is 6 feet. What must be the width of the lower portion?

25. The load P, concentrated at the center, which a homogeneous elliptical plate can support is given by the formula

= 0

where h is the thickness of the plate in., R is the maximum safe unit stress for the material 16,000 pounds per square inch, P = 600π, and m is the ratio of the breadth to the length of the ellipse. Find the value

of m.

26. The maximum stresses on a parabolic arch of a bridge are given by the roots of the equation 2r5 5r4 + 9r2 + 8r + 2 = 0, where r 0, where r is the ratio of the length of the arch occupied by a moving load such as a train. Find r.

54. Graph of the Function f(ax). Before proceeding to the summary in the next section, we shall see how the graph of f(ax) may be found from that of f(x). This will complete the study we shall make of pairs of related functions and their graphs. Consider the

(2x)2 – 6(2x). Show that the graph of f(2x) may be obtained by bisecting the abscissas of points on the graph of ƒ (x).

If we substitute 4 for x in the first function, and half of 4, namely 2, in the second, the results are both equal to – 8. Hence the point (4, -8) lies on the first graph and (2, -8) on the second, and the abscissa of the latter point is half that of the former.

If we substitute any value for x in the first function, and half that value in the second, the results will be the same, namely x2-6x. Hence if (x1, y1) is a point on the first graph the point (x1/2, y1) will be on the second.

-1

y

-2

ƒ(2x) f(x)

As the latter point may be obtained by bisecting the abscissa of the former, it follows that the graph of ƒ(2x) may be obtained by bisecting the abscissas of several points on the graph of ƒ (x) and drawing a smooth curve through them.

The reasoning employed in this example may be used to prove the

Theorem. The graph of f(ax) may be obtained by dividing by a the abscissas of points on the graph of f(x). Corresponding points on the two graphs lie on the same or opposite sides of the y-axis according as a is positive or negative.

X =

This theorem may also be established as follows: Let y = f(x) and suppose that the result of solving this equation for x is x (y). The graphs of these two equations are identical. If we solve the equation y = f(ax) for ax the result must be ax = (y), and the graphs of these two equations are identical. But the last equation may be written x =

α

10(3), from which it follows that the abscissas of points on this curve are one-ath of the abscissas of the points on the graph of (y).

This theorem will find application in the study of some of the transcendental functions. It is also applied in an alternative method of finding the rational zeros of a polynomial in Exercises 2 to 6 below.

If a has such a value as, the division of the abscissas by amounts to multiplication by 3.

55. Related Functions and their Graphs. We have studied several functions which may be obtained by transforming, or changing, a given function, and whose graphs may be obtained from that of the given function by simple geometric constructions or transformations. These are given in the tables below. In order that the statements may be concise and accurate, it is assumed that the constants involved in the table are positive. The changes necessary if the constants are negative should cause no difficulty.

The graph of

may be obtained from the graph of ƒ(x) by

(1)

(2)

(3)

(4)

(5)

f(x)+k moving it up k units (page 19, Exercise 3)
f(x + h) | moving it to the left h units (Theorem 2, page 92)
af (x) | multiplying ordinates by a (Theorem, page 89)
f(ax) | dividing abscissas by a (Theorem, page 151)

1

f(x)

by the principles for the graphs of reciprocal functions in Section 40 (page 117).

the origin

(4b) -ƒ(− x) | (6) inverse of

f(x)

the bisector of the first and third quadrants (Theorem, page 114).

It should be noticed that (3a) and (4a) are the special cases of (3) and (4) respectively, obtained when a = − 1, and that (46) may be regarded as a combination of (3a) and (4a).

Properties (1) and (2) are closely related to the translation of the axes. If we set (Theorem 1, page 89)

in

we get

x = x' + h, y = y' + k

y = f(x),

y' + k = f(x' + h).

The graphs of these two functions are identical if x' and y' are plotted on axes h units to the right and k units above the old axes. But the second equation may be written

y' = f(x' + h) – k;

and hence, if x' and y' are plotted on the old axes the graph of y' may be obtained by moving that of y to the left h units and k units down (properties (1) and (2)).

Properties (3) or (4) may also be interpreted as giving essentially the same distortion to a curve as is obtained if unequal units are chosen on the coördinate axes.

With these properties might be associated the method of ob

taining the graph of y = f(x) g(x) by first plotting the graphs of f(x) and g(x), and then adding or subtracting the corresponding ordinates (see Exercise 4 of the Miscellaneous Exercises following Chapter I).

56. Some Operations of Algebra regarded as Properties of Functions. The identity (a + b)2 = a2 + 2ab+b2

is usually thought of in elementary algebra as a rule for obtaining the square of a binomial. It may also be regarded as a property of the function x2, that is, as the means of expressing the value of the function when x is the sum of two given numbers in terms of the squares and first powers of the separate numbers.

It will be well to group together a few such relations, whether we regard them as rules of operation or as properties of functions. As typical of such relations we select the following properties of the function x":

If f(x) = x2, where n is a positive integer, then

These relations, or special cases of them, are used constantly in transforming algebraic expressions, for example, in simplifying complex fractions.

Relations analogous to some of these will be derived for each of the transcendental functions to be studied in later chapters. In order to perceive the analogy clearly, it is desirable to associate with these relations the general notation in terms of f(x) given at the beginning of each line.

Essential differences between various functions lie in the differences in analogous properties.

A given function may not possess properties analogous to all of the eight relations above. For example, if f(x) = x2,