secting at 0. Taking O as the origin of measurement, or zeropoint, lay off, on any convenient scale, the values of x along X'X, positive to the right, negative to the left, as described in the preceding article. The line X'X we shall call the x-axis.

At each point of the x-axis, erect a perpendicular line of length equal by scale to the magnitude of f(x) corresponding to the value of x at that point, above X'X if the function is positive, below if it is negative; draw a smooth curve through the ends of the perpendiculars. This curve is called the graph of f(x).

The perpendicular distance from any given point of the graph to the x-axis is called the ordinate of the point, and the value of x for the given point is called the abscissa.

The value of f(x) for any given value of x is the length by scale of the ordinate of the curve corresponding to the given value of x. Also, the value of x that causes f(x) to have any given value is the length by scale of the abscissa of that point of the graph whose ordinate is the given value of f(x).

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80. Graph of a Linear Function.-If f(x) is of the first degree, its graph is a straight line, and therefore such a function is called linear.

the values of f(x), for values of x from -4 to +3, are:

X :-4, -3, -2,

– 1,

0, +1, +2, +3

f(x): 0, +1, +1, +3, +2, +1⁄2, +3, +1/2

The points corresponding to these values are plotted in Fig. 3, and the graph drawn through them.

If a straight line is drawn from each point of the graph parallel to the x-axis, as in the figure, it is obvious, from the

successive values of f(x), that the perpendicular in the series of right triangles thus formed is the base. Hence all the right triangles are equal, and the acute angles adjacent to their bases are equal. From the principles of plane geometry this can only be true if the successive portions of the graph lie in the same straight line, AB.

A further deduction may be made from this property, that the increase in f(x) bears a constant ratio to the increase in x. In this particular case the ratio is 1, the coefficient of x in f(x), and in any other linear function (mx+b), the ratio would be m. This ratio measures the tangent of the angle which the straight line makes with the x-axis.

Examples.

Draw the graphs of the following functions:

1. 3x, 2x, x, x, fx.

2. — 2x, −x, −2x+3, −1x+2.

3. Using the same set of axes, draw the graphs of 5x-2 and 3x+4. How do the graphs indicate the value of x for which the two functions are equal?

81. Graph of x2.—We shall find that the graph of any other function of x than the linear function is a curve.

graph of x2. When

Consider the

x= -3, -2, -1, −1, −1, −1, − 1, 0, 1, 1, 1, 1, 1, 2, 3, f(x) = x2= 9, 4, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 4, 9. Plotting these values by the process described, and sketching in a smooth curve through the successive points, we obtain the graph given in Fig. 4.

-3

-2

XI

FIG. 4.

The abscissa and ordinate of any point on this graph represent a number and its square, so that we can determine the square of a number a by measuring the ordinate of the point of the graph having the abscissa a, or we can determine the square root of a number b by measuring the abscissa of the point of the graph having the ordinate b.

Any graph shows the real values of the function corresponding to real values of the variable; this graph shows that every real number has a positive square, that a negative number has no real square root, and that a positive number has two real square roots, numerically equal, but opposite in sign.

The graph also shows that if x starts with a negative value and increases, a2 decreases, rapidly at first, then more and more slowly until it becomes zero when x=0; from here x2 increases as x increases, slowly at first, then more and more rapidly.

82. The Graphs of (x2+a) and (x+a)2.—By properly shifting the axes, we can make the same curve that we drew as the graph of x2 serve as the graph of (x2+a) or of (x+a)2, a being any given number. For instance, suppose we should construct the graph of (x2+1) independently. For any value of x, the point representing (2+1) would be one unit higher than the point representing 2; thus the graph of (x2+1) would be the graph of 2 raised one unit. We can, however, get the same effect more easily by leaving the graph where it is and lowering the x-axis one unit. In the same way, the graph of (x2−2) can be obtained from the graph of x2 by raising the x-axis two units.

Again, suppose we should construct the graph of (x+1)2; in the table of values of x and (x+1)2, we should have the same set of values for (x+1)2 that we had for x2, except that the corresponding value of x would be one less. This means that the graph of (x+1)2 is the graph of x2 shifted one unit to the left, or is the graph of x2 with the y-axis shifted one unit to the right. In the same way, the graph of (x-2)2 can be obtained from the graph of x2 by shifting the y-axis two units to the left.

These considerations are perfectly general, and may be applied to any function:

The graph of f(x)+a can be obtained from the graph of f(x) by shifting the x-axis a units downward; and

The graph of f(x+a) can be obtained from the graph of f(x) by shifting the y-axis a units to the right.

If a is negative, the shift is in the opposite direction.

Examples.

Draw the graphs of the following functions:

1. x2-3. 6. x2+x3.

2. 2x2+1. 3. —†x2. 4. 23. 5. — 2x3.

7. x2 and 2x+3 on the same axes. How do the graphs show the values of x for which these two functions are equal?

8. x3 and 2x-1 on the same axes. How do the graphs show the values of x for which x3-2x+1=0?

83. Slope of the Graph.-The steepness or slope of a graph at any point, which is the same as the slope of the tangent at that point, measures the rapidity with which the function is increasing with respect to its variable. The slope of a straight line is the tangent of the angle made by the line with the axis of x, and is considered positive if the acute angle made above the x-axis by these lines opens to the right, negative if it opens to the left. A decrease is thus regarded as a negative increase. We have already seen the use of the slope in the case of the linear function, mx+b, for which the slope m of the graph showed that when x was increased, mx+b was increased m times as much.

84. Slope of the Graph of x2.-Let P be any point of the graph of x2, having the abscissa x and the ordinate x2. Let PT represent the tangent to the graph at P, and draw PM parallel to OX, of any convenient length; erect a perpendicular to PM at M, meeting PT at T.

MT

The ratio is the desired slope of the graph (or of its

PM

tangent PT) at the point P; in order to find its value, erect a perpendicular to PM at any point N, extending it to meet the graph at Q. Draw the secant PQ and let it meet MT at S.