The reasoning employed above may be used to prove the Theorem. The graph of af(x) may be obtained by multiplying by a the ordinates of points on the graph of f(x). Corresponding points on the two graphs lie on the same or opposite sides of the x-axis according as a is positive or negative.

If a = - 1 the graph is symmetrical to that of f(x) with respect to the x-axis.

This theorem is the second one we have considered belonging to the set of relations between pairs of functions and their graphs (see the last paragraph but one on page 43).

31. Translation of the Coördinate Axes. Consider a system of coördinate axes with the origin O, and a second system, parallel to the first, with origin O'. Replacing the first system by the second is called translating the axes. By the equations for translating the axes we

mean equations which express the coördinates of a point referred to the first, or old, axes in terms of the coördinates of the same point referred to the second, or new, axes.

Let the old and new coördinates of any point P be respectively (x, y) and (x', y'), and let the old coördinates of the new origin, O', be (h, k). From the figure we readily obtain

Y

where (h, k) are the coördinates of the new origin.

(1)

Suppose that the graph of an equation in x and y has been plotted. To find the equation of this graph referred to new axes parallel to the old axes we substitute in the given equation the values of x and y given by (1). The following examples illustrate the utility of the theorem.

EXAMPLE 1. Given the equation

y = x2 - 6x + 13,

(2)

translate the axes so that the new origin is the point (3, 4) and find the equation in the new coördinates which has the same graph as (2). Plot the graph, and state its most noteworthy properties.

FIG. 50.

The graph of (5), plotted on the new axes, is the

same as the graph of (2), referred to the old axes. But the graph of (5) is a parabola which is easily plotted on the new axes. It is shown in the figure. This curve is also the graph of (2) when plotted on the old axes. From it we see that the axis of symmetry of the parabola, the y'-axis, is the line x 3, and that the minimum point is the new origin (3, 4).

In this example it turned out that the equation obtained by translating the axes was much simpler than the given equation. The question arises: If an equation is given, how can we determine how to move the axes so as to obtain a simpler equation with the same graph? The method of doing this is illustrated in

EXAMPLE 2. Simplify the equation

У 2x2 8x + 11

by translating the axes, and construct the graph.

(6)

First solution. Substituting in (6) the values of x and y given by (1),

This equation will be very simple if the coefficient of x' and the constant term are zero, that is, if

and

2h2 - 8h + 11 - k

4h - 8 = 0,
0.

:}

(8)

6 x1

Hence, if the axes be translated so that the origin is moved to the point (2, 3), equation (6) assumes the simpler form (10). Plotting the graph of (10) on the new axes we get the curve in

the figure.

Second solution. The form of the given equation (6) may be changed as follows:

2.

It is seen by inspection of this equation that if we set

which is identical with (10). Equations (11) are the equations for translating the axes so that the origin is moved to the point (2, 3) (by Theorem 1).

Let the graph of any function y = f(x) be given. In order to move the x-axis up k units, leaving the y-axis unchanged, we set y = y' +k, obtaining y' + k = f(x), or

The graph of (12) is identical with the given graph, if the equation is plotted on the new axes.

Now suppose that the graph of (12) is plotted on the original axes. It may be obtained by moving the given graph down k units (see the Theorem in Exercise 3, page 19). Hence the graph of (12) may be interpreted in two ways, which are essentially identical since the effect of moving the x-axis up k

units, and erasing the original x-axis, is the same as moving the graph down k units and erasing the original curve.

In like manner, if we set x = x+h, and leave y unchanged, we get

y = f(x' + h).

(13)

The graph of this equation is identical with the graph of ƒ(x), if plotted on the original x-axis and a new y-axis h units to the right of the old. But the effect of moving the y-axis h units to the right and erasing the old y-axis gives the same figure as moving the curve h units to the left and erasing the old curve. Hence the graph of (13) referred to the original axes may be obtained by moving the graph of f(x) h units to the left.

In plotting (13) on the original axes it is convenient to write x in place of x'. We thus obtain

Theorem 2. The graph of f(x + h) may be obtained by moving the graph of f(x) h units to the left. The motion will be to the right if h is negative.

This theorem, for which we find application in later chapters, should be associated with the last paragraph but one on page 43.

EXERCISES

1. Plot on the same axes the graphs of:

(a) x2, 3x2, x2, — \x2.

(b) x2 4x, 2(x2 - 4x), 1(x2 – 4x).

(c) x2+3, 2(x2 + 3), }(x2 + 3).

3. Show that the points (2, 1), (3, 21), and (4, 4) lie on one of the parabolas ax2.

4. Show that one of the curves ax2 passes through any point P1(x1, Y1).

5. Find the average rate of change of y with respect to x, for the intervals from x 1 to x = 1, from x = 2, from x 2 to x = 3, and for x2; (b) y = 2x2; (c) y = x2; (d) y 1

O to x

any interval if (a) y

=

ax2.

6. Translate the axes to the new origin indicated, and construct both pairs of axes and the graph:

7. Using Theorem 2, Section 31, construct on the same axes the graphs of x2, (x + 2)2, and (x − 3)2.

8. Construct the graphs of x2 and x2 + 8x + 16 on the same axes.

9. On the same axes construct the graphs of y = x2, y = 2x2, y = (x + 2)2, y = x2 + 2.

10. Simplify the following equations by translating the axes. In each case, construct both pairs of axes and the graph. Determine the axis of symmetry and the maximum or minimum point in the given coordinates.

32. Instantaneous Velocity. If a ball is dropped, its velocity changes continually. An approximate value of what we mean when we speak of its velocity at some given instant is given by the average velocity in an interval of time At beginning at that instant. The smaller the value of At is, the more accurate is the approximation. A precise notion of the velocity at an instant is given by the

DEFINITION. The velocity of a body at an instant, or its instantaneous velocity, is the limit* of the average velocity in an interval At beginning at that instant, as the interval At approaches zero.

The computation of an instantaneous velocity is illustrated in the following example.

* The limit of a variable is a constant such that the numerical value of the difference between the variable and the constant becomes and remains less than any assigned positive number, however small.