Transformations of Functions. Shifting Graphs. Similarly, you can obtain the graph of. g x x 2 2 f x 2. Vertical and Horizontal Shifts

Transcription

1 0_007.qd /7/05 : AM Page 7 7 Chapter Functions and Their Graphs.7 Transormations o Functions What ou should learn Use vertical and horizontal shits to sketch graphs o unctions. Use relections to sketch graphs o unctions. Use nonrigid transormations to sketch graphs o unctions. Wh ou should learn it Knowing the graphs o common unctions and knowing how to shit, relect, and stretch graphs o unctions can help ou sketch a wide variet o simple unctions b hand. This skill is useul in sketching graphs o unctions that model real-lie data, such as in Eercise on page, where ou are asked to sketch the graph o a unction that models the amounts o mortgage debt outstanding rom 990 through 00. Shiting Graphs Man unctions have graphs that are simple transormations o the parent graphs summarized in Section.. For eample, ou can obtain the graph o h b shiting the graph o upward two units, as shown in Figure.7. In unction notation, h and are related as ollows. h Similarl, ou can obtain the graph o g Upward shit o two units b shiting the graph o to the right two units, as shown in Figure.77. In this case, the unctions g and have the ollowing relationship. g h() = + Right shit o two units () = g() = ( ) () = FIGURE.7 FIGURE.77 Ken Fisher/Gett Images In items and, be sure ou see that h c corresponds to a right shit and h c corresponds to a let shit or c > 0. The ollowing list summarizes this discussion about horizontal and vertical shits. Vertical and Horizontal Shits Let c be a positive real number. Vertical and horizontal shits in the graph o are represented as ollows.. Vertical shit c units upward: h c. Vertical shit c units downward: h c. Horizontal shit c units to the right: h c. Horizontal shit c units to the let: h c

2 0_007.qd /7/05 : AM Page 75 Section.7 Transormations o Functions 75 You might also wish to illustrate simple transormations o unctions numericall using tables to emphasize what happens to individual ordered pairs. For instance, i ou have, h,and g, ou can illustrate these transormations with the ollowing tables. h g Some graphs can be obtained rom combinations o vertical and horizontal shits, as demonstrated in Eample. Vertical and horizontal shits generate a amil o unctions, each with the same shape but at dierent locations in the plane. Eample Shits in the Graphs o a Function Use the graph o to sketch the graph o each unction. a. g b. h Solution a. Relative to the graph o, the graph o g is a downward shit o one unit, as shown in Figure.7. b. Relative to the graph o, the graph o h involves a let shit o two units and an upward shit o one unit, as shown in Figure.79. () = h() = ( + ) + () = g () = FIGURE.7 FIGURE.79 Now tr Eercise. In Figure.79, notice that the same result is obtained i the vertical shit precedes the horizontal shit or i the horizontal shit precedes the vertical shit. Graphing utilities are ideal tools or eploring translations o unctions. Graph, g, and h in same viewing window. Beore looking at the graphs, tr to predict how the graphs o g and h relate to the graph o. a. b. Eploration, g, h, g, h c., g, h

3 0_007.qd /7/05 : AM Page 7 7 Chapter Functions and Their Graphs () = h() = Relecting Graphs The second common tpe o transormation is a relection. For instance, i ou consider the -ais to be a mirror, the graph o h is the mirror image (or relection) o the graph o, as shown in Figure.0. FIGURE.0 Relections in the Coordinate Aes Relections in the coordinate aes o the graph o are represented as ollows.. Relection in the -ais: h () =. Relection in the -ais: h Eample Finding Equations rom Graphs FIGURE. The graph o the unction given b is shown in Figure.. Each o the graphs in Figure. is a transormation o the graph o. Find an equation or each o these unctions. 5 = g( ) = h( ) Eploration Reverse the order o transormations in Eample. Do ou obtain the same graph? Do the same or Eample. Do ou obtain the same graph? Eplain. FIGURE. Solution a. The graph o g is a relection in the -ais ollowed b an upward shit o two units o the graph o. So, the equation or g is g. b. The graph o h is a horizontal shit o three units to the right ollowed b a relection in the -ais o the graph o. So, the equation or h is h. Now tr Eercise 9.

4 0_007.qd /7/05 : AM Page 77 Section.7 Transormations o Functions 77 Eample Relections and Shits Compare the graph o each unction with the graph o. a. g b. h c. k Algebraic Solution a. The graph o g is a relection o the graph o in the -ais because g. b. The graph o h is a relection o the graph o in the -ais because h. c. The graph o k is a let shit o two units ollowed b a relection in the -ais because k. Graphical Solution a. Graph and g on the same set o coordinate aes. From the graph in Figure., ou can see that the graph o g is a relection o the graph o in the -ais. b. Graph and h on the same set o coordinate aes. From the graph in Figure., ou can see that the graph o h is a relection o the graph o in the -ais. c. Graph and k on the same set o coordinate aes. From the graph in Figure.5, ou can see that the graph o k is a let shit o two units o the graph o, ollowed b a relection in the -ais. () = g() = h() = () = FIGURE. FIGURE. () = k () = + Now tr Eercise 9. FIGURE.5 Activities. How are the graphs o and g related? Answer: The are relections o each other in the -ais.. Compare the graph o with the graph o g 9. Answer: g is shited to the right nine units. When sketching the graphs o unctions involving square roots, remember that the domain must be restricted to eclude negative numbers inside the radical. For instance, here are the domains o the unctions in Eample. Domain o g : 0 Domain o h : 0 Domain o k :

5 0_007.qd /7/05 : AM Page 7 7 Chapter Functions and Their Graphs FIGURE. g() = () = h() = () = FIGURE. () = FIGURE.7 5 g() = h() = Nonrigid Transormations Horizontal shits, vertical shits, and relections are rigid transormations because the basic shape o the graph is unchanged. These transormations change onl the position o the graph in the coordinate plane. Nonrigid transormations are those that cause a distortion a change in the shape o the original graph. For instance, a nonrigid transormation o the graph o is represented b g c, where the transormation is a vertical stretch i c > and a vertical shrink i 0 < c <. Another nonrigid transormation o the graph o is represented b h c, where the transormation is a horizontal shrink i c > and a horizontal stretch i 0 < c <. Eample Nonrigid Transormations Compare the graph o each unction with the graph o h a. b. Solution a. Relative to the graph o, the graph o h is a vertical stretch (each -value is multiplied b ) o the graph o. (See Figure..) b. Similarl, the graph o g g is a vertical shrink each -value is multiplied b o the graph o. (See Figure.7.) Now tr Eercise. Eample 5 Nonrigid Transormations Compare the graph o each unction with the graph o. a. g b. h Solution a. Relative to the graph o, the graph o g. is a horizontal shrink c > o the graph o. (See Figure..) b. Similarl, the graph o () = FIGURE.9 h is a horizontal stretch 0 < c < o the graph o. (See Figure.9.) Now tr Eercise 7.

6 0_007.qd /7/05 : AM Page 79 Section.7 Transormations o Functions 79.7 Eercises VOCABULARY CHECK: In Eercises 5, ill in the blanks.. Horizontal shits, vertical shits, and relections are called transormations.. A relection in the -ais o is represented b h, while a relection in the -ais o is represented b h.. Transormations that cause a distortion in the shape o the graph o are called transormations.. A nonrigid transormation o represented b h c is a i c > and a i 0 < c <. 5. A nonrigid transormation o represented b g c is a i c > and a i 0 < c <.. Match the rigid transormation o with the correct representation o the graph o h, where c > 0. h c (i) A horizontal shit o, c units to the right h c (ii) A vertical shit o, c units downward (c) h c (iii) A horizontal shit o, c units to the let (d) h c (iv) A vertical shit o, c units upward PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed or this section at For each unction, sketch (on the same set o coordinate aes) a graph o each unction or c,, and. c c (c) c. For each unction, sketch (on the same set o coordinate aes) a graph o each unction or c,,, and. c c (c) c. For each unction, sketch (on the same set o coordinate aes) a graph o each unction or c, 0, and. c c (c) c. For each unction, sketch (on the same set o coordinate aes) a graph o each unction or c,,, and. c, c, c, c, < 0 0 < 0 0 In Eercises 5, use the graph o to sketch each graph.to print an enlarged cop o the graph go to the website (c) (c) (d) (d) (e) (e) () () (g) (g) (, 0) (, ) (, ) (0, ) (, ) (, ) (, ) (0, ) FIGURE FOR 5 FIGURE FOR (c) (c) (d) (d) (e) (e) () () (g) 0 (g)

7 0_007.qd /7/05 : AM Page 0 0 Chapter Functions and Their Graphs (, ) (0, ) (, 0) (, ) (0, 5) (, 0) (, 0) 0 (, ) (, ) 0. Use the graph o to write an equation or each FIGURE FOR 7 FIGURE FOR 9. Use the graph o to write an equation or each (c) 0. Use the graph o to write an equation or each (c) (d) (d) (c). Use the graph o to write an equation or each (c) In Eercises, identi the parent unction and the transormation shown in the graph. Write an equation or the unction shown in the graph. (d) (d)

8 0_007.qd /7/05 : AM Page Section.7 Transormations o Functions In Eercises 9, g is related to one o the parent unctions described in this chapter. Identi the parent unction. Describe the sequence o tranormations rom to g. (c) Sketch the graph o g. (d) Use unction notation to write g in terms o. 9. g 0. g. g 7. g. g. g 7 5. g 5. g g. g 9. g 0. g 0. g. g 5. g. g 9 5. g. g 5 7. g 9. g 9. g 7 0. g. g. g In Eercises 50, write an equation or the unction that is described b the given characteristics.. The shape o, but moved two units to the right and eight units downward. The shape o, but moved three units to the let, seven units upward, and relected in the -ais 5. The shape o, but moved units to the right. The shape o, but moved si units to the let, si units downward, and relected in the -ais 7. The shape o, but moved 0 units upward and relected in the -ais. The shape o, but moved one unit to the let and seven units downward 9. The shape o, but moved si units to the let and relected in both the -ais and the -ais 50. The shape o, but moved nine units downward and relected in both the -ais and the -ais 5. Use the graph o to write an equation or each 5. Use the graph o to write an equation or each 5. Use the graph o to write an equation or each 5. Use the graph o to write an equation or each 0 5 (, ) (, ) (, ) (, ) 0 (, ) (, 7) (, ) (, )

9 0_007.qd /7/05 : AM Page Chapter Functions and Their Graphs In Eercises 55 0, identi the parent unction and the transormation shown in the graph. Write an equation or the unction shown in the graph. Then use a graphing utilit to veri our answer Graphical Analsis In Eercises, use the viewing window shown to write a possible equation or the transormation o the parent unction Graphical Reasoning In Eercises 5 and, use the graph o to sketch the graph o g. To print an enlarged cop o the graph, go to the website g (c) g g (e) g g 5 g (c) g (d) g (e) g () g 0 5 g (d) () g 7. Fuel Use The amounts o uel F (in billions o gallons) used b trucks rom 90 through 00 can be approimated b the unction F t t, 0 t where t represents the ear, with t 0 corresponding to 90. (Source: U.S. Federal Highwa Administration) Describe the transormation o the parent unction. Then sketch the graph over the speciied domain. Find the average rate o change o the unction rom 90 to 00. Interpret our answer in the contet o the problem. (c) Rewrite the unction so that t 0 represents 990. Eplain how ou got our answer. (d) Use the model rom part (c) to predict the amount o uel used b trucks in 00. Does our answer seem reasonable? Eplain. Model It

10 0_007.qd /7/05 : PM Page Section.7 Transormations o Functions. Finance The amounts M (in trillions o dollars) o mortgage debt outstanding in the United States rom 990 through 00 can be approimated b the unction M t t 0.9, where t represents the ear, with t 0 corresponding to 990. (Source: Board o Governors o the Federal Reserve Sstem) Describe the transormation o the parent unction. Then sketch the graph over the speciied domain. Rewrite the unction so that t 0 represents 000. Eplain how ou got our answer. Snthesis True or False? In Eercises 9 and 70, determine whether the statement is true or alse. Justi our answer. 9. The graphs o and are identical. 70. I the graph o the parent unction is moved si units to the right, three units upward, and relected in the -ais, then the point, 9 will lie on the graph o the transormation. 7. Describing Proits Management originall predicted that the proits rom the sales o a new product would be approimated b the graph o the unction shown. The actual proits are shown b the unction g along with a verbal description. Use the concepts o transormations o graphs to write g in terms o. 0,000 0,000 The proits were onl three-ourths as large as epected. The proits were consistentl $0,000 greater than predicted. 0 t t 0,000 0,000 0,000 0,000 g g t t (c) There was a two-ear dela in the introduction o the product. Ater sales began, proits grew as epected. 7. Eplain wh the graph o is a relection o the graph o about the -ais. 7. The graph o passes through the points 0,,,, and,. Find the corresponding points on the graph o. 7. Think About It You can use either o two methods to graph a unction: plotting points or translating a parent unction as shown in this section. Which method o graphing do ou preer to use or each unction? Eplain. Skills Review In Eercises 75, perorm the operation and simpli ,000 0,000 In Eercises and, evaluate the unction at the speciied values o the independent variable and simpli.. (c). 0 0 (c) 0 In Eercises 5, ind the domain o the unction g t