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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:

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

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(11)

we will obtain the simpler equation

y' = 2x22,

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

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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.

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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

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2. Find the value of a if the graph of y = ax2 passes through the point

(2, 3), and construct the graph.

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 = 0 to x = 1, from x = 1 to x = 2, from x = 2 to x = 3, and for any interval if (a) y = x2; (b) y 2x2; (c) y = x2; (d) y 6. Translate the axes to the pairs of axes and the graph:

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=

=

ax2. new origin indicated, and construct both

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

y

=

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, 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.

(a) x2 + 6x + 4.
(c) 2x2 16x + 35.

(e) - x2 + 2x + 5.

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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.

EXAMPLE. If a ball is dropped, its distance from the starting point at any time t, in seconds, is given by

Find the velocity at any time.,

8 = 16t2.

(1)

The distance s at any time t is given by (1). If the distance changes by As in the next At seconds, the distance at the time t + At will be s + As. These corresponding values of the distance and time must satisfy (1), and hence

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This is the average velocity during an interval of At seconds beginning at the time t. By the definition above, the velocity at the time t is the limit of (3) as At approaches zero. Denoting it by v, we get

v =

32t,

since 16At approaches zero when At approaches zero.

(4)

33. Rate of Change. Slope of Tangent Line. A generalization of the idea of instantaneous velocity, which may be described as the rate of change of the distance s with respect to the time t, is given in the

DEFINITION. The rate of change of y with respect to x for a given value of x is the limit, as Ax approaches zero, of the average rate of change of y with respect to x in the interval from x to x + Ax.

If

y is a function of x,

y = f(x),

(1)

the rate of change of y with respect to x may be determined as follows:

Substitute x + Ax for x and y + Ay for y in (1). The result,

y + Ay = f(x + Ax),

(2)

is true because Ay is the change in y due to a change of Ax in x, and hence x + Ax and y + Ay are corresponding values of the independent variable and the function.

Subtract (1) from (2), which gives Ay, and divide by Ax, which gives Ay/Ax, the average rate of change of y in the interval Ax. Then find the limit of the average rate of change as Ax approaches zero. To interpret this process graphically we shall need the

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P

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FIG. 53.

-T

DEFINITION. If P is a given point on a curve and Q any other point on it, the line tangent to the curve at P is the limiting position of the secant PQ as Q moves along the curve and approaches P. The points P(x, y) and Q(x + Ax, y + ▲y) are on the graph of (1), and ▲y/Ax is the slope of the secant PQ. As Ax approaches zero, Ay/Ax approaches the rate of change of y; also, Qapproaches P, the secant PQ approaches the tangent PT, and the slope of PQ approaches that of PT. Hence,

The rate of change of y with respect to x is represented by the slope of a line tangent to the graph of y.

The tangent to the graph of y at a maximum or minimum point is horizontal, and hence its slope is zero, and hence, also the rate of change of y.

This fact affords a general method of finding maximum and minimum points.

The slope of the tangent line and the rate of change of the function should be added to the table of properties of graphs and functions on page 42.

EXAMPLE 1. Find the rate of change of

y = ax2,

and discuss its meaning for the graph.

Replacing x and y by x + Ax and y + Ay respectively, we get

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which is the average rate of change of y in any interval ▲x.

(3)

(4)

(5)

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