Transformations and Graphs

Elementary Transformations

The graphs of many functions can be regarded as arising from more basic graphs as a result of one or more elementary transformations. The elementary transformations considered here are shifting, stretching and compression, and reflection with respect to a coordinate axis.

Basic Function

Given a basic function $ y = f(x) $ with the graph shown in Fig. 11-1, the following transformations have easily identified effects on the graph.

Figure 11-1

V ertical Shifting

The graph of $ y = f(x) + k $ , for k > 0, is the same as the graph of $ y = f(x) $ shifted up k units. The graph of $ y = f(x) + k $ , for k < 0, is the same as the graph of $ y = f(x) $ shifted down k units.

EXAMPLE 11.1 For the basic function shown in Fig. 11-1, graph $ y = f(x) $ and $ y = f(x) + 2 $ on the same coordinate system (Fig. 11-2), $ y = f(x) $ and $ y = f(x) - 2.5 $ on the same coordinate system (Fig. 11-3).

Figure 11-2

Figure 11-3

V ertical Stretching and Compression

The graph of $ y = af(x) $ , for a > 1, is the same as the graph of $ y = f(x) $ stretched, with respect to the y-axis, by a factor of a. The graph of $ y = af(x) $ , for 0 < a < 1, is the same as the graph of $ y = f(x) $ compressed, with respect to the y-axis, by a factor of 1/a.

EXAMPLE 11.2 For the basic function shown in Fig. 11-1, graph $ y = f(x) $ and $ y = 2f(x) $ on the same coordinate system (Fig. 11-4); $ y = f(x) $ and $ y = f(x) $ on the same coordinate system (Fig. 11-5).

Figure 11-4

Figure 11-5

Horizontal Shifting

The graph of \(y = f(x + h)\), for \(h > 0\), is the same as the graph of \(y = f(x)\) shifted left h units. The graph of \(y = f(x - h)\), for \(h > 0\), is the same as the graph of \(y = f(x)\) shifted right h units.

EXAMPLE 11.3 For the basic function shown in Fig. 11-1, graph $ y = f(x) $ and $ y = f(x + 2) $ on the same coordinate system (Fig. 11-6); $ y = f(x) $ and $ y = f(x - 1) $ on the same coordinate system (Fig. 11-7).

Figure 11-6

Figure 11-7

Horizontal Stretching and Compression

The graph of $ y = f(ax) $ , for a > 1, is the same as the graph of $ y = f(x) $ compressed, with respect to the x-axis, by a factor of a. The graph of $ y = f(ax) $ , for 0 < a < 1, is the same as the graph of $ y = f(x) $ stretched, with respect to the x-axis, by a factor of 1/a.

EXAMPLE 11.4 For the basic function shown in Fig. 11-1, graph $ y = f(x) $ and $ y = f(2x) $ on the same coordinate system (Fig 11-8); $ y = f(x) $ and $ y = f(x) $ on the same coordinate system (Fig. 11-9).

Figure 11-8

Figure 11-9

Reflection with Respect to a Coordinate Axis

The graph of $ y = -f(x) $ is the same as the graph of $ y = f(x) $ reflected across the x-axis. The graph of $ y = f(-x) $ is the same as the graph of $ y = f(x) $ reflected across the y-axis.

EXAMPLE 11.5 For the basic function shown in Fig. 11-1, graph $ y = f(x) $ and $ y = -f(x) $ on the same coordinate system (Fig. 11-10); $ y = f(x) $ and $ y = f(-x) $ on the same coordinate system (Fig. 11-11).

Figure 11-10

Figure 11-11

SOLVED PROBLEMS

11.1. Explain why, for positive h, the graph of $ y = f(x) + h $ is displaced up by h units from the graph of $ y = f(x) $ , while the graph of $ y = f(x + h) $ is displaced left by h units.

Consider the point $ (a,f(a)) $ on the graph of $ y = f(x) $ . The point $ (a, f(a) + h) $ on the graph of $ y = f(x) + h $ can be regarded as the corresponding point. This point has y-coordinate h units more than that of the original point $ (a, f(a)) $ , and thus has been displaced up h units.

It is not helpful to regard the point $ (a, f(a + h)) $ as the corresponding point on the graph of $ y = f(x + h) $ . Rather, consider the point with x-coordinate a - h; thus y-coordinate $ f(a - h + h) = f(a) $ . Then the point $ (a - h, f(a)) $ is easily seen to have x-coordinate h units less than that of the original point $ (a, f(a)) $ ; thus, it has been displaced left h units.

11.2. Explain why the graph of an even function is unchanged by a reflection with respect to the y-axis.

A reflection with respect to the y-axis replaces the graph of $ y = f(x) $ with the graph of $ y = f(-x) $ . Since for an even function $ f(-x) = f(x) $ , the graph of an even function is unchanged.

11.3. Explain why the graph of an odd function is altered in exactly the same way by reflection with respect to the x-axis or the y-axis.

A reflection with respect to the x-axis replaces the graph of $ y = f(x) $ with the graph of $ y = -f(x) $ , while a reflection with respect to the y-axis replaces the graph of $ y = f(x) $ with the graph of $ y = f(-x) $ . Since for an odd function $ f(-x) = -f(x) $ , the two reflections have exactly the same effect.

11.4. Given the graph of y = lxl as shown in Fig. 11-12, sketch the graphs of (a) y = lxl−1; (b) y = lx − 2l;

  1. $ y = |x + 2| - 1 $ ; (d) $ y = -2|x| + 3 $ .

Figure 11-12
  1. The graph of $ y = | x | - 1 $ (Fig. 11-13) is the same as the graph of $ y = | x | $ shifted down 1 unit.

Figure 11-13

  1. The graph of $ y = x + 2 - 1 $ (Fig. 11-15) is the same as the graph of $ y = x $ shifted left by 2 units and then down 1 unit.

  2. The graph of $ y = x - 2(x) $ is the same as the graph of $ y = x $ shifted right by 2 units.

Figure 11-14

Figure 11-15
  1. The graph of $ y = -2|x| + 3 $ (Fig. 11-16) is the same as the graph of $ y = |x| $ stretched by a factor of 2, reflected with respect to the x-axis, and shifted up 3 units.

Figure 11-16

11.5. Given the graph of $ y = $ as shown in Fig. 11-17, sketch the graphs of (a) $ y = $ ; (b) $ y = -3 $ ; (c) $ y = $ ; (d) $ y = -1.5 + 2 $ .

Figure 11-17
  1. The graph of $ y = $ (Fig. 11-18) is the same as the graph of $ y = $ reflected with respect to the y-axis.

Figure 11-18
  1. The graph of $ y = -3 $ (Fig. 11-19) is the same as the graph of $ y = $ stretched by a factor of 3 with respect to the y-axis and reflected with respect to the x-axis.

Figure 11-19
  1. The graph of $ y = $ (Fig. 11-20) is the same as the graph of $ y = $ shifted left 3 units and compressed by a factor of 2 with respect to the y-axis.

Figure 11-20
  1. The graph of $ y = -1.5 + 2 $ (Fig. 11-21) is the same as the graph of $ y = $ shifted right 1 unit, stretched by a factor of 1.5 and reflected with respect to the x-axis, and shifted up 2 units.

Figure 11-21

11.6. Given the graph of $ y = x^{3} $ as shown in Fig. 11-22, sketch the graphs of (a) $ y = 4 - x^{3} $ ; (b) $ y = (x)^{3} - $ .

Figure 11-22
  1. The graph of $ y = 4 - x^{3} $ (Fig. 11-23) is the same as the graph of $ y = x^{3} $ reflected with respect to the x-axis and shifted up 4 units.

Figure 11-23
  1. The graph of $ y = ( x )^{3} - $ (Fig. 11-24) is the same as the graph of $ y = x^{3} $ stretched by a factor of 2 with respect to the x-axis and shifted down $ $ unit.

Figure 11-24

SUPPLEMENTARY PROBLEMS

11.7. Given the graph of $ y = x^{1/3} $ as shown in Fig. 11-25, sketch the graphs of:

\[ \begin{array}{r l r}&{\mathrm{(a)~}y=2x^{1/3}+1;\mathrm{(b)~}y=2(x+1)^{1/3};\mathrm{(c)~}y=2-x^{1/3};\mathrm{(d)~}y=(-2x)^{1/3}-1.}\end{array} \]

Figure 11-25

Ans. (a) See Fig. 11-26; (b) see Fig. 11-27; (c) see Fig. 11-28; (d) see Fig. 11-29.

Figure 11-26

Figure 11-27

Figure 11-28

Figure 11-29

11.8. (a) Describe how the graph of $ y = |f(x)| $ is related to the graph of $ y = f(x) $ . (b) Given the graph of $ y = x^{2} $ as shown in Fig. 11-30, sketch first the graph of $ y = x^{2} - 4 $ , then the graph of $ y = |x^{2} - 4| $ .

Figure 11-30

Ans. (a) The portions of the graph above the x-axis are identical to the original, while the portions of the graph below the x-axis are reflected with respect to the x-axis.

  1. See Figs. 11-31 and 11-32.

Figure 11-31

Figure 11-32

11.9. (a) Describe how the graph of $ x = f(y) $ is related to the graph of $ y = f(x) $ . (b) Given the graphs shown in the previous problem, sketch the graphs of $ x = y^{2} $ and $ x = |y^{2} - 4| $ .

Ans. (a) The graph is reflected with respect to the line y = x.

  1. See Figs. 11-33 and 11-34.

Figure 11-33

Figure 11-34