2. Find the perimeter of an equilateral triangle as a function of the

altitude x.

3. Express the area of a square as a function of (a) the side, (b) the diagonal.

4. Express the area of a regular hexagon as a function of the side.

5. A regular octagon is formed by cutting off isosceles right triangles from the corners of a square. Express the area as a function of the side of one of the right triangles cut off.

6. Express the side of a regular decagon as a function of the radius of the circumscribed circle.

7. A Norman window consists of a square surmounted by a semicircle. Find the area as a function of the side of the square x.

8. (a) A man walks from a certain town toward a second at the rate of three miles an hour. Express the distance traveled as a function of the time.

(b) A second man starts at the same time from the second town, which is 10 miles from the first, and travels at the rate of 4 miles an hour toward the first town. Express his distance from the first town as a function of the time.

(c) A third man starts from the same town as the first man but two hours later and travels at the rate of 31⁄2 miles an hour in the same direction. Express his distance from town as a function of the time elapsed since the first man started.

How far will each man be from the first town 4 hours after the first man starts? When will each man be 20 miles from the first town?

9. A man starts from a town 15 miles directly west of a city and travels east at the rate of 4 miles an hour. A second man starts from the same town at the same time and travels west at the rate of 3 miles an hour. Express the distance of each from the city after t hours as a function of t. How far from the city will each be in 3 hours? When will they be 25 miles apart?

10. A ball starting from rest rolls down an inclined plane 4 feet in the first second, 8 feet in the next, 12 feet in the next, etc. Express the distance it rolls in any second t as a function of t. How far will it roll in the 8th second?

11. With the data of the preceding problem and the formula for the sum of an arithmetic progression find the total distance rolled as a function of the time t. When will the ball have rolled 108 feet?

12. A body starting from rest falls 16 feet in the first second, 48 feet in the next, 80 feet in the next, etc. Find the distance fallen in any second as a function of the time of falling. Find the total distance fallen as a function of the time.

13. $100 is placed at simple interest at 4 per cent. Express the amount at the end of t years as a function of t.

14. If the interest in the preceding problem is compounded annually, express the amount at the end of t years as a function of t.

15. The cross-section of a gutter pipe is in the form of an isosceles trapezoid. The lower base and the inclined sides are each 3 inches long. Find the area as a function of the width across the top.

16. A rectangle is inscribed in a circle 4 inches in diameter. Express the area as a function of one of the sides.

7. Graphical Representation. Directed Lines. The functional relation can be represented by a geometrical figure which furnishes a valuable method for studying the properties of the function, since the whole of the relation is placed before the mind at once. The system of coördinates devised by René Descartes (1596-1650), which is developed in the following section, is the basis of the representation. This system of coördinates rests on the theory of directed lines.

Let XX' be any line, and let the direction from X' to X be called positive, from X to X' negative. These words are used instead of such terms as north and south, to the right and left, up and down, backward and forward. A line upon which a positive direction has been fixed is called a directed line. The positive direction is commonly indicated by an arrow-head. If A and B be two points on a directed line, the symbol AB is used to denote either:

(1) The line drawn from A to B, or

(2) The real number whose numerical value is the number of times the unit of length is contained in the line, and whose sign is positive or negative according as the direction from A to B is positive or negative.

Thus, in the figure, AB and A'B' denote certain lines.

They also denote the numbers AB = 3 and A'B' = -2, provided the unit of length is a quarter of an inch.

It follows that if A and B are points on a directed line

DEFINITION. No matter what the position of three points A, B, C, on a directed line may be, the sum of AB and BC is defined to be

If AB is thought of as a motion from A to B, and BC as a motion from B to C, the sum gives a motion from A to C. The sum gives the distance from A to C in both magnitude and direction, but not necessarily the total number of units passed over in going from A to B and then from B to C.

That this definition agrees with elementary algebra, when AB, BC, and AC are regarded as numbers, is seen in the following illustrations, of which the first agrees with arithmetic.

An especially important use of directed lines is the following: Let O be any point on a directed line X'X, and let points be laid off on each side of O at a unit's distance from each other. Then with every point P on X'X is associated a real number OP, and conversely, with every real number is associated a point on the line.

In the figure, the numbers associated with the points A, B, C, D are respectively OA 3, OB 3, OC 5, OD -4.5.

=

= 1

=

=

Using this association of points and numbers, if P1 and P2 are two points on the line, and x and x2 are the numbers associated with them, we have from (2)

This difference of the values of the x's is denoted by Ax, so

that

Ax = P1 P2 = X2 X1

(4)

Notice that Ax is a single symbol (never the product of two numbers ▲ and x), and that its value is obtained by subtracting the value of x corresponding to the first point from that corresponding to the second.

EXERCISES

1. Illustrate (3) by numerical examples for the six possible relative positions of O, P1, and P2. Find Ax for each case.

2. Show that Ax is positive or negative according as P1 lies to the left or right of P2, using the definition that a <b if b - a is positive.

3. If OP = x, show that x increases or decreases according as P moves to the right or left.

8. Rectangular Coördinates. Let X'X and Y'Y be two

are called rectangular coördinates of P, x the abscissa, and y the

ordinate. In the figure, the ordinate y is often thought of as MP, which equals ON in both magnitude and direction. The directed lines X'X and Y' Y are called the axes of coördinates and their intersection O the origin.

The abscissa of a point is positive or negative according as the point lies to the right or left of the y-axis. The ordinate is positive or negative according as the point lies above or below the x-axis. In the figure, the abscissa x is positive, while the ordinate y is negative.

As seen above, any point P determines a pair of real numbers, its coördinates. Conversely, given any pair of real numbers, x and y, a point may be plotted, that is, constructed, whose coördinates are x and y. For on X'X lay off OM = x. At M erect a line perpendicular to X'X and on it lay off MP = y. Then P is the required point.

The symbol (x, y) is used to mean the point whose coördinates are x and y. If P is this point, it is indicated by the symbol P(x, y).

Coördinate axes divide the plane into four parts called quadrants. These are numbered as in the figure, which also indicates the signs of the coördinates of a point in every quadrant.

We are frequently concerned with points which are symmetric with respect to the origin, the axes, or a line bisecting the first and third quadrants. Points having these symmetric relations are determined in accordance with the

DEFINITIONS. (1) Two points are symmetric with respect to a

ΥΛ

II (-,-+)

I (+,+)

X'

X

III (-,-)

IV (+;-)

Y'

FIG. 13.

third point if the line joining the two points is bisected at the third point.

(2) Two points are symmetric with respect to a line, if the line is the perpendicular bisector of the line joining the two points.