STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Axis Combining Transformations of Graphs


 Wesley Harmon
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1 6 CHAPTER Analsis of Graphs of Functions. STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Ais Combining Transformations of Graphs In the previous section, we saw how adding or subtracting a constant can cause a vertical or horizontal shift. Now we will see how multipling b a constant alters the graph of a function. Vertical Stretching TECHNOLOGY NOTE B defining Y as directed in parts A, B, and C, and defining Y, Y, and Y 4 as shown here, ou can minimize our kestrokes. (These graphs will not appear unless Y is defined.) = f() = c f(), c > FIGURE 8 FOR DISCUSSION In each group, we give four related functions. Graph the four functions in the first group (Group A), and then answer the questions regarding those functions. Repeat the process for Group B and Group C. Use the window specified for each group. A B C 5, 5 b 5, 5, 5 b 5,, b, How does the graph of compare to the graph of?. How does the graph of compare to the graph of?. How does the graph of 4 compare to the graph of? 4. If we choose c 4, how do ou think the graph of 5 c would compare to the graph of 4? Provide support b choosing such a value of c. In each group of functions in the preceding activit, we started with a basic function and observed how the graphs of functions of the form c compared with the graph of for positive values of c that began at and became progressivel larger. In each case, we obtained a vertical stretch of the graph of the basic function with which we started. These observations can be generalized to an function. Vertical Stretching of the Graph of a Function If c, the graph of c f is obtained b verticall stretching the graph of f b a factor of c. In general, the larger the value of c, the greater the stretch. In Figure 8, we graphicall interpret the statement above. EXAMPLE Recognizing Vertical Stretches Figure 9 shows the graphs of four functions. The graph labeled is that of the function defined b f. The other three functions,,, and 4, are defined as follows, but not necessaril in the given order:.4,., and 4..
2 . Stretching, Shrinking, and Reflecting Graphs 7 Determine the correct equation for each graph. 4 5 = f() = FIGURE 9 Solution The values of c here are.4,., and 4.. The vertical heights of the points with the same coordinates of the three graphs will correspond to the magnitudes of these c values. Thus, the graph just above will be that of.4, the highest graph will be that of 4., and the graph of. will lie between the others. Therefore, If we were to trace to an point on the graph of and then move to the other graphs one b one, we would see that the values of the points would be multiplied b the appropriate values of c. You ma wish to eperiment with our calculator in this wa. Vertical Shrinking 4.,.4, and 4.. TECHNOLOGY NOTE You can use a screen such as this to minimize our kestrokes in parts A, B, and C. Again, Y must be defined in order to obtain the other graphs. FOR DISCUSSION This discussion parallels the one given earlier in this section. Follow the same general directions. (Note: The fractions 4,, and 4 ma be entered as their decimal equivalents when plotting the graphs.) A B C 5, 5 b 5, 5, 5 b, 5, b, How does the graph of compare to the graph of?. How does the graph of compare to the graph of?. How does the graph of 4 compare to the graph of? 4. If we choose c 4, how do ou think the graph of 5 c would compare to the graph of 4? Provide support b choosing such a value of c. In this For Discussion activit, we began with a basic function and observed the graphs of c, as we chose progressivel smaller values of c, with c. In each case, the graph of was verticall shrunk (or compressed). These observations can also be generalized to an function.
3 8 CHAPTER Analsis of Graphs of Functions = f() Vertical Shrinking of the Graph of a Function If c, the graph of c f is obtained b verticall shrinking the graph of f b a factor of c. In general, the smaller the value of c, the greater the shrink. = c f(), < c < FIGURE TECHNOLOGY NOTE Figure shows a graphical interpretation of vertical shrinking. EXAMPLE Recognizing Vertical Shrinks Figure shows the graphs of four functions. The graph labeled is that of the function defined b f. The other three functions,,, and 4, are defined as follows, but not necessaril in the given order:.5,., and.. Determine the correct equation for each graph. = f() = 4 This method of defining Y and Y using a list of coefficients in Y will allow ou to duplicate Figure. TECHNOLOGY NOTE B defining Y as directed in parts A, B, C, and D, and defining Y as shown here, ou can minimize our kestrokes FIGURE Solution The smaller the positive value of c, where c, the more compressed toward the ais the graph will be. Since we have c.5,., and., the function rules must be as follows: Reflecting Across an Ais.,.5, and 4.. We have seen how graphs can be transformed b shifting, stretching, and shrinking. We now eamine how graphs can be reflected across an ais. FOR DISCUSSION In each pair, we give two related functions. Graph f and f in the standard viewing window, and then answer the questions below for each pair. A B C D With respect to the ais,. how does the graph of compare to the graph of?. how would the graph of compare with the graph of, based on our answer to Item? Confirm our answer b graphing.
4 . Stretching, Shrinking, and Reflecting Graphs 9 TECHNOLOGY NOTE B defining Y as directed in parts E, F, and G, and defining Y as shown here (using function notation), ou can minimize our kestrokes. Again, in each pair, we give two related functions. Graph f and f in the standard viewing window, and then answer the questions below for each pair. E F G 4 4 With respect to the ais,. how does the graph of compare to the graph of? 4. how would the graph of compare with the graph of, based on our answer to Item? Confirm our answer b graphing. The results of the preceding discussion can be formall summarized. Reflecting the Graph of a Function Across an Ais For a function defined b f, (a) the graph of f is a reflection of the graph of f across the ais. (b) the graph of f is a reflection of the graph of f across the ais. Figure shows how the reflections just described affect the graph of a function in general. (, 6) (7, 6) = f() ( 4, ) (, ) FIGURE (, ) ( 4, ) = f() (, 6) (7, 6) FIGURE 4 = f () = f () = f ( ) b (a, b) b ( a, b) (a, b) a a a (a, b) b = f () Reflection across the ais Reflection across the ais (a) (b) FIGURE EXAMPLE Appling Reflections across Aes Figure shows the graph of a function f. (a) Sketch the graph of f. (b) Sketch the graph of f. Solution (a) We must reflect the graph across the ais. This means that if a point a, b lies on the graph of f, then the point a, b must lie on the graph of f. Using the labeled points, we find the graph of f in Figure 4.
5 CHAPTER Analsis of Graphs of Functions ( 7, 6) (, 6) = f( ) (4, ) (, ) FIGURE 5 (b) Here we must reflect the graph across the ais, meaning that if a point a, b lies on the graph of f, then the point a, b must lie on the graph of f. Thus, we obtain the graph of f as shown in Figure 5. To illustrate reflections on calculatorgenerated graphs, observe Figure 6. Figure 6(a) shows that Y has been defined b 6 and Y Y, which means that the graph of Y is a reflection across the ais. Figure 6(b) shows the graphs of Y and Y, confirming this fact. Notice that Y Y, indicating that the graph of is a reflection across the ais. This is confirmed b Figure 6(c). Y Y = Y = Y = Y ( ) Y = Y (a) (b) FIGURE 6 (c) What Went WRONG? To see how negative values of a affect the graph of a, a student entered three functions Y, Y, and Y as in the accompaning screen. The calculator graphed the first two as shown, but gave a snta error when it attempted to graph the third. What Went Wrong? graph for? What must the student do in order to obtain the desired Answers to What Went Wrong? The student used a subtraction sign to define Y rather than a negative sign. Notice the difference between the signs in Y and as compared to Y. The student must reenter using a negative sign. Y Y
6 . Stretching, Shrinking, and Reflecting Graphs = = FIGURE 7 Combining Transformations of Graphs The graphs of and are shown in the same viewing window in Figure 7. In terms of the tpes of transformations we have studied, the graph of is obtained b verticall stretching the graph of b a factor of and then reflecting across the ais. Thus, we have a combination of transformations. As ou might epect, we can create an infinite number of functions b verticall stretching or shrinking, shifting upward, downward, left, or right, and reflecting across an ais. The net eample investigates eamples of this tpe of function. In determining the order in which the transformations are made, use the order of operations. EXAMPLE 4 Describing a Combination of Transformations of a Graph (a) Describe how the graph of 4 5 can be obtained b transforming the graph of. Illustrate with a graphing calculator. (b) Give the equation of the function that would be obtained b starting with the graph of, shifting units to the left, verticall shrinking the graph b a factor of, reflecting across the ais, and shifting the graph 4 units downward, in this order. Illustrate with a graphing calculator. Analtic Solution (a) The presence of 4 in the definition of the function indicates that the graph of must be shifted 4 units to the right. Since the coefficient of 4 is (a negative number with absolute value greater than ), the graph is stretched verticall b a factor of and then reflected across the ais. Finall, the constant 5 indicates that the graph is shifted upward 5 units. These ideas are summarized below. ➂ Reflect across the ais. ➁ Stretch b a factor of. ➀ Shift 4 units to the right. 4 5 ➃ Shift 5 units upward. (b) Shifting units to the left means that is transformed to. Verticall shrinking b a factor of means multipling b, and reflecting across the ais changes to. Finall, shifting 4 units downward means subtracting 4. Putting this all together leads to the equation 4. Graphing Calculator Solution (a) Figure 8 supports the discussion in the analtic solution. FIGURE 8 (b) Figure 9 supports the discussion in the analtic solution. = ( 4) + 5 = + 4 FIGURE 9 = =
7 CHAPTER Analsis of Graphs of Functions CAUTION The order in which the transformations are made is important. If the are made in a different order, a different equation can result. See the diagram that follows. ➀ Stretch b a factor of. ➁ Shift units upward. ➀ Shift units to the left. ➁ Stretch b a factor of. EXAMPLE 5 Recognizing a Combination of Transformations Figure 4 shows two views of the graph of and another graph illustrating a combination of transformations. Find the equation of the transformed graph.. =. = (a). (b) FIGURE 4 Solution Figure 4(a) shows that the lowest point on the transformed graph has coordinates,, indicating that the graph has been shifted units to the right and units downward. Figure 4(b) shows that a point on the right side of the transformed graph has coordinates 4,, and thus the slope of this ra is m 4 Thus, the stretch factor is. This information leads to as the equation of the transformed graph... EXERCISES Write the equation that results in the desired transformation.. The squaring function, stretched b a factor of. The square root function, reflected across the ais. The cubing function, shrunk b a factor of 4. The cube root function, reflected across the ais
8 . Stretching, Shrinking, and Reflecting Graphs 5. The absolute value function, stretched b a factor of and reflected across the ais 7. The cubing function, shrunk b a factor of.5 and reflected across the ais 6. The absolute value function, shrunk b a factor of and reflected across the ais 8. The square root function, shrunk b a factor of. and reflected across the ais Use the concepts of this chapter to draw a rough sketch of the graphs of,, and. Do not plot points. In each case, and can be graphed b one or more of these: a vertical and/or horizontal shift of the graph of, a vertical stretch or shrink of the graph of, or a reflection of the graph of across an ais. After ou have made our sketches, check b graphing them in an appropriate viewing window of our calculator. 9.., 4,,, 4.,,.., 6, ,, 6.,,,,.5,, 7.,, 4 8. Concept Check Suppose that the graph of f is smmetric with respect to the ais and it is reflected across the ais. How will the new graph compare with the original one? Fill in each blank with the appropriate response. (Remember that the vertical stretch or shrink factor is positive.) 9. The graph of 4 can be obtained from the graph of b verticall stretching b a factor of and reflecting across the ais.. The graph of 6 can be obtained from the graph of b verticall stretching b a factor of and reflecting across the ais.. The graph of 4 can be obtained from the graph of b shifting horizontall units to the, verticall shrinking b a factor of, reflecting across the ais, and shifting verticall units in the direction.. The graph of 5 6 can be obtained from the graph of b reflecting across the ais, verticall shrinking b a factor of, reflecting across the ais, and shifting verticall units in the direction.. The graph of 6 can be obtained from the graph of b shifting horizontall units to the and stretching verticall b a factor of. 4. The graph of.5 can be obtained from the graph of b shifting horizontall units to the and shrinking verticall b a factor of. Give the equation of each function whose graph is described. 5. The graph of is verticall shrunk b a factor of and the resulting graph is shifted 7 units downward. 6. The graph of is verticall stretched b a factor of. This graph is then reflected across the ais. Finall, the graph is shifted 8 units upward. 7. The graph of is shifted units to the right. This graph is then verticall stretched b a factor of 4.5. Finall, the graph is shifted 6 units downward. 8. The graph of is shifted units to the left. This graph is then verticall stretched b a factor of.5. Finall, the graph is shifted 8 units upward.,
9 4 CHAPTER Analsis of Graphs of Functions Shown on the left is the graph of Y in the standard viewing window of a graphing calculator. Si other functions, Y through Y 7, are graphed according to the rules shown in the screen on the right. Y = ( ) + Match each function with its calculatorgenerated graph from choices A F first without using a calculator, b appling the techniques of this chapter. Then, confirm our answer b graphing the function on our calculator Y Y Y 4 Y 5 Y 6 Y 7 A. B. C. D. E. F. In Eercises 5 and 6, the graph of f has been transformed to the graph of g. No shrinking or stretching is involved. Give the equation of g = f() = = g() (5, ) (4, ) = g() = f() =
10 . Stretching, Shrinking, and Reflecting Graphs 5 In Eercises 7 4, each figure shows the graph of a function f. Sketch b hand the graphs of the functions in parts (a), (b), and (c), and answer the question of part (d). 7. (a) f (b) f (c) f 8. (a) f (b) f (c) f (d) What is f? (d) What is f 4? 9. (a) f (b) f (c) f 4. (a) f (b) f (c) f (d) What are the intercepts of f? (d) On what interval of the domain is f? (, ) (, ) = f() (6, ) (4, ) = f() (.5,.5) (, ) (, ) (, ) (,.5) (, ) ( 4, ) (, ) = f() (, ) (6, ) (, ) = f() 4. (a) f (b) f (c).5f 4. (a) f (b) f (c) f (d) What smmetr does the graph of f ehibit? (d) What smmetr does the graph of f ehibit? = f() (, ) (, ) (, ) (, ) (, ) (, ) 4. Concept Check If r is an intercept of the graph of f, what statement can be made about the intercept of the graph of each of the following? (Hint: Make a sketch.) (a) f (b) f (c) f (, ) = f () (, ) 44. Concept Check If b is the intercept of the graph of f, what statement can be made about the intercept of the graph of each of the following? (Hint: Make a sketch.) (a) f (b) f (c) 5f (d) f Concept Check The sketch shows an eample of a function defined b f that increases on the interval a, b. Use this graph as a visual aid, and appl the concepts of reflection introduced in this section to answer each question. (Make our own sketch if ou wish.) = f() 45. Does the graph of f increase or decrease on the interval a, b? 46. Does the graph of f increase or decrease on the interval b, a? 47. Does the graph of f increase or decrease on the interval b, a? a b 48. If c, does the graph of c f increase or decrease on the interval a, b?
11 6 CHAPTER Analsis of Graphs of Functions State the intervals over which each function is (a) increasing, (b) decreasing, and (c) constant. 49. The function graphed in Figure 5. The function graphed in Figure 4 5. The function graphed in Figure (See Figure 9.) In Eercises 5 55, each function has a graph with an endpoint (a translation of the point, ). Enter each into our calculator in an appropriate viewing window, and using our knowledge of the graph of, determine the domain and range of the function. (Hint: Locate the endpoint.) Concept Check Based on our observations in Eercise 5, what are the domain and range of f a h k, if a, h, and k? Concept Check Shown here are the graphs of and 5. The point whose coordinates are given at the bottom of the screen lies on the graph of. Use this graph, not our calculator, to find the value of for the same value of shown = 5 = = = Reviewing Basic Concepts (Sections..). Suppose that f is defined for all real numbers, and f 6. For the given assumptions, find another function value. (a) The graph of f is smmetric with respect to the origin. (b) The graph of f is smmetric with respect to the ais. (c) For all, f f. (d) For all, f f.. Match each equation in Column I with a description of its graph from Column II as it relates to the graph. I II (a) 7 A. a shift of 7 units to the left (b) 7 B. a shift of 7 units to the right (c) 7 C. a shift of 7 units upward (d) 7 D. a shift of 7 units downward (e) 7 E. a vertical stretch b a factor of 7
12 . Stretching, Shrinking, and Reflecting Graphs 7. Match each equation in parts (a) (h) with the sketch of its graph. The basic graph,, is shown here. (a) (b) (c) (d) = 4 (e) (f) (g) (h) A. B. C. D. 4 4 E. F. G. H (, ) (, ) 4. Match each equation with its calculatorgenerated graph. (a) 6 (b) 6 (c) (d) 4 (e) 4 6 A. B. C. D. E.
13 8 CHAPTER Analsis of Graphs of Functions 5. Each graph is obtained from the graph of f or g b appling the transformations discussed in Sections. and.. Describe the transformations, and then give the equation for the graph. (a) (b) Suppose F is changed to F h. How are the graphs of these equations related? Is the graph of F h the same as the graph of F h? If not, how do the differ? 8. Suppose the equation F is changed to c F, for some constant c. What is the effect on the graph of F? Discuss the effect depending on whether c or c, and c or c. 9. Complete the table if (a) f is an even function and (b) f is an odd function. (c) 5 (d) 6. Consider the two functions in the figure. (a) Find a value of c for which g f c. (b) Find a value of c for which g f c. 7 = g() f() (Modeling) Carbon Monoide Levels The 8hour maimum carbon monoide levels (in parts per million) for the United States from 98 to 99 can be modeled b the function defined b f , where corresponds to 98. (Source: U.S. Environmental Protection Agenc, 99.) Find a function represented b g that models the same carbon monoide levels ecept that is the actual ear between 98 and 99. For eample, g 985 f and g 99 f 8. (Hint: Use a horizontal translation.) = f()
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