8. What is the abscissa of any point on the Y axis?

9. The coördinates of a variable point are restricted so that its ordinate is always 2. Where may the point move?

10. If both ordinate and abscissa of a point vanish, can the point move? Where will it be?

11. Plot the quadrilateral whose vertices are (0, 0), (− 1, — 8). What kind of a quadrilateral is it?

6,3), (5, — 5),

12. The coördinates of three vertices of a parallelogram are (— 1, − 1), (6, 2), (− 1, −6). Find the coördinates of the fourth vertex.

13. The coördinates of two adjacent vertices of a square are (— 1, − 2) and (1, 2). Find the coördinates of the remaining vertices (two solutions). Plot the figures.

14. The coördinates of two adjacent vertices of a rectangle are (-1,-2), (1, 2). What restriction is imposed on the coördinates of remaining vertices?

15. The coördinates of the extremities of the bases of an isosceles triangle are (1, 6), (1, 2). Where may the vertex lie? What restriction is imposed on the coördinates (x, y) of the vertex?

(2,1)

101. The graph of an equation. The equation x = 2y is satisfied by numberless pairs of values (x, y); for example, (2, 1), (0, 0), (1, 1), (— 2, − 1) all satisfy the equation. There are, however, numberless pairs of values which do not satisfy the equation; for example, (1, 2), (2, -1), (-1,1), (0, 1). The pairs of values which satisfy the equation may be taken as the coördinates of points in a plane. The totality of such points would thus in a sense represent the equation, for it would serve to distinguish the points whose coördinates do satisfy the equation from those whose coördinates do not. After finding a few pairs of values which satisfy the above equation we note that any point whose abscissa is twice the ordinate, i.e. for which x = 2y, is a point whose coördinates satisfy the equation. Any such point lies on the straight line through the origin and the point (2, 1). We can then say that those points

(1,2)

(-11)

(1%)

(0,0)

2,-1)

(0,-1)

|(2,-1)

and only those which are on the straight line represented in the figure have coördinates which satisfy the equation. This line is the graphical representation or graph of the equation.

The equation of a line or a curve is satisfied by the coördinates of every point on that line or curve.

Any point whose coördinates satisfy an equation is on the graph of the equation.

102. Restriction to coördinates. In § 100 it was seen that a moving point whose coördinates were unrestricted took on every position in the plane. We now see that when the coördinates of a point are restricted so as to satisfy a certain equation (as x = 2y), the motion of the point is no longer free, but restrained to move along a certain path. Thus, for instance, the equation x = 4 means that the path of the moving point is so restricted that its abscissa is always 4. Its ordinate is still unrestricted and may have any value. This shows that the plot of x 4 is a straight line four units to the right of the Y axis and parallel to it, for the abscissa of every point on that line is 4, and every point whose abscissa is 4 lies on that line.

=

EXERCISES

Determine on what line the moving point is restricted to move by the following equations. Draw the

graph.

103. Plotting equations. Plotting an equation consists in finding the line or curve the coördinates of whose points satisfy the equation. Thus the process of § 101 was nothing else than plotting the equation x = 2 y. This may be done in some cases by observing what restriction the equation imposes on the coördinates of the moving point; but more often we are obliged to form a

table of various solutions of the equation, and to form a curve by joining the points corresponding to these solutions. This gives us merely an approximate figure of the exact graph which becomes more accurate as we find the coördinates of points closer to each other on the line or curve.

RULE. When y is alone on one side of the equation, set x equal to convenient integers and compute the corresponding values of y. Arrange the results in tabular form. Take corresponding values of x and y as coördinates and plot the various points. Join adjacent points, making the entire plot a smooth curve. When x is alone on one side of the equation integral values of y may be assumed and the corresponding values of x computed.

Care should be taken to join the points in the proper order so that the resulting curve pictures the variation of y when x increases continuously through the values assumed for it. By adjacent points we mean points corresponding to adjacent values of x.

Any scale of units along the X and Y axes that is convenient may be adopted. The scales should be so chosen that the portion of the curve that shows considerable curvature may be displayed in its relation to the axes and the origin.

When there is any question regarding the position of the curve between two integral values of x, an intermediate fractional value of x may be substituted, the corresponding value of y found, and thus an additional point obtained to fix the position of the curve in the vicinity in question.

2. y = x2 - 7x + 1.

4. y = x2 - 3x + 2.

6. y=x3- 2x + 1.

8. y = 2x2 - 6x + 7.

10. y = 2x2-6x – 3.

3. y = x2 + 1.

5. y=x3- 4x.

7. y = x2 + 6x + 5.

9. y=2x2 - 3x + 4.

11. y = x2 - 12x + 11.

104. Plotting equations after solution. When neither x nor y is already alone on one side of the equation, the equation should be solved for y (or x) and the rule of the previous section applied. It should be noted that when a root is extracted two values of y may correspond to a single value of x.

Assuming the various integral values for x, we obtain the following table and plot:

YA

In this example, when x is greater than 3 or less than Thus none of the curves is found outside a strip x =

3, y is imaginary.

= ± 3.

To find exactly where the curve crosses the X axis, the equation may be solved for x and the value of x corresponding to y = 0 found. Thus

If y = 0, x = ± √2.1. This point is included in the plot.

15. 6x2+2x+3y2=0. 16. x2 + 2 x + 1 = y2 − 3 y.

105. Graph of the linear equation. The intimate relation between the simplest equations and the simplest curves is given in the following theorems.

THEOREM I. The graph of the equation y = ax is the straight line through the origin and the point (1, a), where a is number.

The proof falls into two parts.

any

real

First. Any point on the line through the origin and the point (1, a) has coördinates that satisfy the equation. Let P (Figure 1) with coördinates (x', y') be on the line 04. By similar triangles