Section 1.3: Transformations of Graphs
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1 CHAPTER 1 A Review of Functions Section 1.3: Transformations of Graphs Vertical and Horizontal Shifts of Graphs Reflecting, Stretching, and Shrinking of Graphs Combining Transformations Vertical and Horizontal Shifts of Graphs Graphs of Some Basic Functions: 54
2 SECTION 1.3 Transformations of Graphs MATH 1330 Precalculus 55
3 CHAPTER 1 A Review of Functions 56
4 SECTION 1.3 Transformations of Graphs MATH 1330 Precalculus 57
5 CHAPTER 1 A Review of Functions Vertical Shifts of Graphs: Horizontal Shifts of Graphs: 58
6 SECTION 1.3 Transformations of Graphs Eample: Solution: MATH 1330 Precalculus 59
7 CHAPTER 1 A Review of Functions Eample: Solution: Additional Eample 1: 60
8 SECTION 1.3 Transformations of Graphs Solution: Additional Eample : MATH 1330 Precalculus 61
9 CHAPTER 1 A Review of Functions Solution: 6
10 SECTION 1.3 Transformations of Graphs Additional Eample 3: Solution: MATH 1330 Precalculus 63
11 CHAPTER 1 A Review of Functions Additional Eample 4: Solution: 64
12 SECTION 1.3 Transformations of Graphs MATH 1330 Precalculus 65
13 CHAPTER 1 A Review of Functions Additional Eample 5: Solution: 66
14 SECTION 1.3 Transformations of Graphs MATH 1330 Precalculus 67
15 CHAPTER 1 A Review of Functions Reflecting, Stretching, and Shrinking of Graphs Reflecting Graphs: 68
16 SECTION 1.3 Transformations of Graphs Stretching and Shrinking of Graphs: MATH 1330 Precalculus 69
17 CHAPTER 1 A Review of Functions 70
18 SECTION 1.3 Transformations of Graphs Eample: Solution: Eample: Solution: MATH 1330 Precalculus 71
19 CHAPTER 1 A Review of Functions Eample: Solution: Eample: 7
20 SECTION 1.3 Transformations of Graphs Solution: Solution 1: MATH 1330 Precalculus 73
21 CHAPTER 1 A Review of Functions 74
22 SECTION 1.3 Transformations of Graphs Solution : MATH 1330 Precalculus 75
23 CHAPTER 1 A Review of Functions 76
24 SECTION 1.3 Transformations of Graphs Additional Eample 1: Solution: MATH 1330 Precalculus 77
25 CHAPTER 1 A Review of Functions Additional Eample : Solution: 78
26 SECTION 1.3 Transformations of Graphs Additional Eample 3: Solution: MATH 1330 Precalculus 79
27 CHAPTER 1 A Review of Functions 80
28 SECTION 1.3 Transformations of Graphs Combining Transformations Suppose that you want to graph the function f 3 7. We can quickly identify from the function that the base function is g, and that there has been a vertical stretch with a factor of 3, a shift left of units, and a downward shift of 7 units. If you are graphing this function, does the order matter when you perform the transformations? For eample, can you shift down, then do the vertical stretch, then shift left? Or should you first shift left, then shift down, and then perform the vertical stretch? We could come up with many different possibilities for the order of transformations for this problem. In this particular eample, the order does matter, and we could get an incorrect graph if we perform certain operations out of order. (There are other cases where the order does not matter, depending on which transformations are used.) It is worth spending some time analyzing the order of transformations which can be done algebraically, without any trial-and-error in graphing. First, remember the rules for transformations of functions. (These are not listed in any recommended order; they are just listed for review.) RULES FOR TRANSFORMATIONS OF FUNCTIONS If f is the original function, a 0 and c 0 : Function Transformation of the graph of f () f c Shift f f c Shift f f c f c Shift f Shift f f Reflect f f Reflect f a f, a 1 Stretch a f, 0 a 1 Shrink upward c units downward c units to the left c units to the right c units in the -ais in the y-ais f vertically by a factor of a. f vertically by a factor of a. f a, a 1 Shrink f horizontally by a factor of 1 a. f a, 0 a 1 Stretch f horizontally by a factor of 1 a. MATH 1330 Precalculus 81
29 CHAPTER 1 A Review of Functions Let us look at Eamples 1 through 6 below, and we will then look for a pattern as to when the order of transformations matters. Eample Problem 1: Start with the function f, and write the function which results from the given transformations. Then decide if the results from parts (a) and (b) are equivalent. (a) Shift upward 7 units, then right units. (b) Shift right units, then upward 7 units. (c) Do parts (a) and (b) yield the same function? (You should be able to tell without graphing.) Solution: (a) f g 7 h 7 Up 7 Right (b) f g h 7 Right Up 7 Yes, parts (a) and (b) yield the same function. Eample Problem : Start with the function f, and write the function which results from the given transformations. Then decide if the results from parts (a) and (b) are equivalent. (a) Stretch vertically by a factor of, then shift downward 5 units. (b) Shift downward 5 units, then stretch vertically by a factor of. Do parts (a) and (b) yield the same function? (You should be able to tell without graphing.) Solution: (a) f g h 5 Stretch vertically by a factor of Down 5 8
30 (b) f g 5 h 5 Down 5 Stretch vertically by a factor of SECTION 1.3 Transformations of Graphs Note: In part (b), h can also be written as h 10. No, parts (a) and (b) do not yield the same function, since Both graphs are shown below to emphasize the difference in the final results (but we can see that the above functions are different without graphing the functions). y y 5 h h 5 10 Eample Problem 3: Start with the function f, and write the function which results from the given transformations. Then decide if the results from parts (a) and (b) are equivalent. (a) Reflect in the y-ais, then shift upward 6 units. (b) Shift upward 6 units, then reflect in the y-ais. Do parts (a) and (b) yield the same function? (You should be able to tell without graphing.) Solution: (a) f g h 6 Reflect in the y-ais Up 6 MATH 1330 Precalculus 83
31 CHAPTER 1 A Review of Functions (b) f g 6 h 6 Up 6 Reflect in the y-ais (c) Yes, parts (a) and (b) yield the same function. Eample Problem 4: Start with the function f, and write the function which results from the given transformations. Then decide if the results from parts (a) and (b) are equivalent. (a) Reflect in the y-ais, then shift left units. (b) Shift left units, then reflect in the y-ais. (c) Do parts (a) and (b) yield the same function? (You should be able to tell without graphing.) Solution: (a) f g h Note: In part (a), Reflect in the y-ais Left h can also be written as h. (b) f g h Left Reflect in the y-ais (c) No, parts (a) and (b) do not yield the same function, since. Both graphs are shown below to emphasize the difference in the final results (but we can see that the above functions are different without graphing the functions). Part (a): h y Part (b): h y 84
32 SECTION 1.3 Transformations of Graphs Eample Problem 5: Start with the function f, and write the function which results from the given transformations. Then decide if the results from parts (a) and (b) are equivalent. (a) Reflect in the -ais, then shift upward 4 units. (b) Shift upward 4 units, then reflect in the -ais. (c) Do parts (a) and (b) yield the same function? (You should be able to tell without graphing.) Solution: (a) f g h 4 Reflect in the -ais Up 4 (b) f g 4 h 4 Up 4 Reflect in the -ais Note: In part (b), h can also be written as h 4. (c) No, parts (a) and (b) do not yield the same function, since 4 4. Both graphs are shown below to emphasize the difference in the final results (but we can see that the above functions are different without graphing the functions). Part (a): h 4 Part (b): h 4 4 y y MATH 1330 Precalculus 85
33 CHAPTER 1 A Review of Functions Eample Problem 6: Start with the function f, and write the function which results from the given transformations. Then decide if the results from parts (a) and (b) are equivalent. (a) Shrink horizontally by a factor of 1, then shift right 6 units. 3 (b) Shift right 6 units, then shrink horizontally by a factor of 1 3. (c) Do parts (a) and (b) yield the same function? (You should be able to tell without graphing.) Solution: (a) f g 3 h 3 6 Note: In part (a), Shrink horizontally Right 6 by a factor of 1 3 h can also be written as h 3 18 (b) f g h Right 6 Shrink horizontally by a factor of (c) No, parts (a) and (b) do not yield the same function, since Both graphs are shown below to emphasize the difference in the final results (but we can see that the above functions are different without graphing the functions). 1 3 Part (a): h y Part (b): h 3 6 y 86
34 SECTION 1.3 Transformations of Graphs Looking for a Pattern When Does the Order of Transformations Matter? When deciding whether the order of the transformations matters, it helps to think about whether a transformation affects the graph vertically (i.e. changes the y-values) or horizontally (i.e. changes the -values). Transformation Shifting up or down Shifting left or right Reflecting in the y-ais Reflecting in the -ais Vertical stretching/shrinking Horizontal stretching/shrinking Vertical or Horizontal Effect? Vertical Horizontal Horizontal Vertical Vertical Horizontal A summary of the results from Eamples 1 through 6 are below, along with whether or not each transformation had a vertical or horizontal effect on the graph. E 1 E E 3 E 4 E 5 E 6 Summary of Results from Eamples 1 6 with notations about the vertical or horizontal effect on the graph, where V = Vertical effect on graph H = Horizontal effect on graph First Set of Transformations (with notations about horizontal/vertical effect) (a) Up 7 (V) Right (H) (a) Vertical stretch, factor of (V) Down 5 (V) (a) Reflect in y-ais (H) Up 6 (V) (a) Reflect in y-ais (H) Left (H) (a) Reflect in -ais (V) Up 4 (V) (a) Horizontal shrink, factor of 1 3 (H) Right 6 (H) Second Set of Transformations (with notations about horizontal/vertical effect) (b) Right (H) Up 7 (V) (b) Down 5 (V) Vertical stretch, factor of (V) (b) Up 6 (V) Reflect in y-ais (H) (b) Left (H) Reflect in y-ais (H) (b) Up 4 (V) Reflect in -ais (V) (b) Right 6 (H) Horizontal shrink, factor of 1 3 (H) Did (a) and (b) yield the same function? The functions were the same. The functions were NOT the same. The functions were the same. The functions were NOT the same. The functions were NOT the same. The functions were NOT the same MATH 1330 Precalculus 87
35 CHAPTER 1 A Review of Functions Notice that in eamples 1 and 3, the order of the transformations did not matter. In both of those eamples, one of the transformations had a vertical effect on the graph, and the other transformation had a horizontal effect on the graph. In eamples, 4, 5 and 6, the order of the transformations did matter. Notice that eample had two vertically-oriented transformations, eample 4 had two horizontallyoriented transformations, eample 5 had two vertically-oriented transformations, and eample 6 had two horizontally-oriented transformations. When you perform two or more transformations that have a vertical effect on the graph, the order of those transformations may affect the final results. Similarly, when you perform two or more transformations that have a horizontal effect on the graph, the order of those transformations may affect the final results. The vertically-oriented transformations do not affect the horizontally-oriented transformations, and vice versa. Let us now return to the function used at the start of this discussion: Eample Problem 7: Suppose that you want to graph f 3 7. In what order can you perform the transformations to obtain the correct graph? Solution: First, decide on the transformations that need to be performed on f 3 7 (without consideration of correct order). Make a note of whether each transformation has a horizontal or vertical effect on the graph. f 3 7 f 3 7 f 3 7 Vertical Stretch, factor of 3 Shift left Shift down 7 (Vertical Effect) ( Horizontal Effect) (Vertical Effect) Notice that the shift to the left is the only transformation that has a horizontal effect on the graph. This transformation can be performed at any point in the graphing process. We need to be more careful about the order in which we perform the vertical stretch and the downward shift, since they both have a vertical effect on the graph. Perform the following transformations algebraically on g to see which one gives the desired function, f 3 7. (The shift left is written first, but we could put that transformation at any point in the process and get the same result.) 88
36 SECTION 1.3 Transformations of Graphs Choice 1: Shift left units, then stretch vertically by a factor of 3, then shift downward 7 units: g h k 3 f 3 7 Left Vertical stretch, factor of 3 Down 7 Choice : Shift left units, then shift downward 7 units, then stretch vertically by a factor of 3: g h k f Left Down 7 Vertical stretch, factor of 3 Notice that our final result for Choice can be written as f 3 1, which is not the desired function. We can see from the analysis above that Choice 1 yields the desired function for Eample Problem 7, which means that in this particular eample, the vertical stretch needs to be performed before the downward shift. Since the left shift can be performed at any point in the process, any of the following order of transformations would yield the correct graph: Shift left units, then stretch vertically by a factor of 3, then shift downward 7 units. Stretch vertically by a factor of 3, then shift left units, then shift downward 7 units. Stretch vertically by a factor of 3, then shift downward 7 units, then shift left units. In this course, you would only need to give one of the answers from the above list (not all three). This eplanation is given to help you understand that there can be multiple solutions for a given problem and how to determine an acceptable order of transformations for the given problem. Eample Problem 8: Suppose that you want to graph f 7. In what order can you perform the transformations to obtain the correct graph? Solution: First, decide on the transformations that need to be performed on f 7 (without consideration of correct order). Make a note of whether each transformation has a horizontal or vertical effect on the graph. MATH 1330 Precalculus 89
37 CHAPTER 1 A Review of Functions 7 7 f f f 7 Reflect in the y-ais Shift left Shift down 7 (Horizontal Effect) (Horizontal Effect) (Vertical Effect) Notice that the downward shift is the only transformation that has a vertical effect on the graph. This transformation can be performed at any point in the graphing process. We need to be more careful about the order in which we perform the reflection in the y- ais and the shift to the left, since they both have a horizontal effect on the graph. Perform the following transformations algebraically on to see which one g gives the desired function, f 7. (The downward shift is written first, but we could put that transformation at any point in the process and get the same result.) Choice 1: Downward shift of 7 units, then reflect in the y-ais, then shift left units. 7 g h k 7 f 7 Down 7 Reflect in the Shift left units y-ais Notice that our final result for Choice 1 can be written as f 7, which is not the desired function. Choice : Downward shift of 7 units, then shift left units, then reflect in the y-ais. 7 f g h k 7 7 Down 7 Shift left units Reflect in the y-ais We can see from the analysis above that Choice yields the desired function for Eample Problem 8, which means that in this particular eample, the shift to the left needs to be performed before the reflection in the y-ais. Since the downward shift can be performed at any point in the process, any of the following order of transformations would yield the correct graph: Shift downward 7 units, then shift left units, then reflect in the y-ais. Shift left units, then shift downward 7 units, then reflect in the y-ais. Shift left units, then reflect in the y-ais, then shift downward 7 units. As mentioned in Eample Problem 7, you would only need to give one of the answers from the above list (not all three). This eplanation is given to help you understand that 90
38 SECTION 1.3 Transformations of Graphs there can be multiple solutions for a given problem and how to determine an acceptable order of transformations for the given problem. Alternate Analysis of Eample Problem 8: It is possible to use a different set of transformations to yield the same result in Eample Problem 8. Remember that the desired function is f 7. You may notice that f 7 7. Looking at f 7, notice that the from the original function is now, which indicates a shift right of units (rather than a shift left of units). The order of the downward shift again does not matter, but we need to decide if we should reflect in the y-ais and then shift right, or shift right and then reflect in the y-ais. Choice 1: Shift downward 7 units (can be done at any time), reflect in the y-ais, then shift right units: 7 7 g h k f Down 7 Reflect in the Shift right units y-ais Notice that our final result for Choice 1 can be written as f 7, which is the desired function. 7 Choice : Shift downward 7 units (can be done at any time), shift right units, then reflect in the y- ais. 7 f g h k 7 7 Down 7 Shift right units Notice that Choice does not yield the desired function. Reflect in the y-ais We can see from the alternate analysis above that Choice 1 yields the desired function for Eample Problem 8, which means that in this particular eample, the reflection in the y- ais needs to be performed before the shift to the right. Since the downward shift can be performed at any point in the process, any of the following order of transformations would yield the correct graph: Shift downward 7 units, then reflect in the y-ais, then shift right units. Reflect in the y-ais, then shift downward 7 units, then shift right units. Reflect in the y-ais, then shift right units, then shift downward 7 units. MATH 1330 Precalculus 91
39 CHAPTER 1 A Review of Functions Remember that when we used a shift to the left instead, we obtained the following solutions as well: Shift downward 7 units, then shift left units, then reflect in the y-ais. Shift left units, then shift downward 7 units, then reflect in the y-ais. Shift left units, then reflect in the y-ais, then shift downward 7 units. In all of the eamples above, we have discussed problems where there were two or three transformations. What if there are four or more transformations? In that case, look at the vertically-oriented transformations and decide the order in which they need to be done. Look at the horizontally-oriented transformations and decide the order in which they need to be done. Then remember that the vertically-oriented transformations have no effect on the horizontally-oriented transformations, and vice versa. Eample Problem 9: Suppose that you want to graph f 3 7. (a) In what order can you perform the transformations to obtain the correct graph? (b) Graph the function. Solution: (a) In order to save time, we have chosen an eample that is similar to Eample Problems 7 and 8. We have the following transformations, not necessarily to be performed in this order: Stretch vertically, factor of 3 Reflection in the y-ais Shift left units* Shift downward 7 units Vertically-oriented transformation Horizontally-oriented transformation Horizontally-oriented transformation Vertically-oriented transformation *For simplicity, we are choosing to focus on a left shift for this problem, instead of factoring out a negative under the radical, where f 3 7, and focusing on a right shift instead. As seen in Eample Problem 8, we would obtain additional solutions if we considered a right shift as an alternate means of solving this problem. (The point of these detailed eplanations is not for the student to be able to list all possible orders of transformations, but to be able to determine one order of transformations for any given problem which would yield a correct graph.) Let us look first at the vertically-oriented transformations. (Temporarily put aside the horizontally-oriented transformations, and just look at what is happening outside the radical sign.) As in Eample Problem 7, the vertical stretch needs to be performed before the downward shift. Net look at the horizontally-oriented transformations. (Temporarily put aside the vertically-oriented transformations, and just look at what is happening under the 9
40 SECTION 1.3 Transformations of Graphs radical sign.) As in Eample Problem 8, the shift to the left needs to be performed before the reflection in the y-ais. Remember that the vertically-oriented transformations do not affect the horizontallyoriented transformations, and vice versa. There are many correct solutions to this problem. Just be sure that in your answer, the vertical stretch is performed before the downward shift, and the shift to the left is performed before the reflection in the y- ais. All the solutions involving a left shift are shown below. You do not need to find all of the answers below; any one of the solutions below would be acceptable. (Note: If we looked at all the possibilities involving a right shift, we would add si more solutions which would yield the same graph.) Stretch vertically by a factor of 3, then shift downward 7 units, then shift left units, then reflect in the y-ais. Shift left units, then reflect in the y-ais, then stretch vertically by a factor of 3, then shift downward 7 units. Stretch vertically by a factor of 3, then shift left units, then shift downward 7 units, then reflect in the y-ais. Stretch vertically by a factor of 3, then shift left units, then reflect in the y-ais, then shift downward 7 units,. Shift left units, then stretch vertically by a factor of 3, then reflect in the y-ais, then shift downward 7 units. Shift left units, then stretch vertically by a factor of 3, then shift downward 7 units, then reflect in the y-ais. (b) To graph the function, we can choose any one of the solutions shown above (and obtain the same result). We will perform the transformations in the order listed for the first solution listed above: Starting with the base graph y : Stretch vertically by a factor of 3, then shift downward 7 units, then shift left units, then reflect in the y-ais. All steps are shown consecutively below. MATH 1330 Precalculus 93
41 CHAPTER 1 A Review of Functions Step 1: y Step : y 3 (Stretch vertically by a factor of 3) y y Step 3: y 3 7 (Shift downward 7 units) Step 4: y 3 7 (Shift left units) y y Step 5: y f 3 7 (Reflect in the y-ais) y Step 5 represents the final result. 94
42 SECTION 1.3 Transformations of Graphs Additional Eample 1: Sketch the graph of the function f 1 apply transformations to the graph of a standard function.. Do not plot points, but instead Analysis of the Order of Transformations for Additional Eample 1: In order to graph this problem correctly, we need to choose a correct order of transformations. First, decide on the transformations that need to be performed on f (without consideration of correct order). Make a note of whether each 1 transformation has a horizontal or vertical effect on the graph. 1 f 1 f 1 f Reflect in the -ais Stretch vertically, factor of Shift up 1 (Vertical Effect) (Vertical Effect) (Vertical Effect) Remember that any vertically-oriented transformations have no effect on horizontallyoriented transformations, and vice versa. However, the order in which you perform vertically-oriented transformations may make a difference in the graph, and the order in which you perform horizontally-oriented transformations may make a difference in the graph. In this eample, notice that all three transformations have a vertical effect on the graph, so the order in which we choose the transformations may matter. You do NOT need to evaluate all the possible combinations in which the above transformations can be performed, but you need to find one which algebraically yields the correct function (and therefore also yields the correct graph). For the purpose of being complete in our solution, we have shown all possible combinations of transformations below so that you can evaluate whether or not your order of transformations is correct. Choice 1: Reflect in the -ais, stretch vertically by a factor of, shift up 1 g h k f 1 Choice 1 yields the desired function, f 1. Choice : Reflect in the -ais, shift up 1, stretch vertically by a factor of. g h k 1 f 1 Choice does NOT yield the desired function, f 1. Choice 3: Stretch vertically by a factor of, reflect in the -ais, shift up 1 g h k f 1 Choice 3 yields the desired function, f 1. MATH 1330 Precalculus 95
43 CHAPTER 1 A Review of Functions Choice 4: Stretch vertically by a factor of, shift up 1, reflect in the -ais g h k 1 f 1 1 Choice 4 does NOT yield the desired function, f 1. Choice 5: Shift up 1, reflect in the -ais, stretch vertically by a factor of. g h 1 k 1 f 1 Choice 5 does NOT yield the desired function, f 1. Choice 6: Shift up 1, stretch vertically by a factor of, reflect in the -ais. g h 1 k 1 f 1 Choice 6 does NOT yield the desired function, f 1. We can see from the si choices above that the following combinations of transformations yield the desired function (and that all the rest will yield an incorrect result): Choice 1: Reflect in the -ais, stretch vertically by a factor of, shift up 1 Choice 3: Stretch vertically by a factor of, reflect in the -ais, shift up 1 If you were working this problem, you could stop analyzing choices as soon as you find one that works and then proceed to graph the function. The solution shown below uses the order of transformations shown in Choice 3. Solution to Additional Eample 1: 96
44 SECTION 1.3 Transformations of Graphs MATH 1330 Precalculus 97
45 CHAPTER 1 A Review of Functions 98
46 SECTION 1.3 Transformations of Graphs Additional Eample : Sketch the graph of the function f 3. Do not plot points, but instead apply transformations to the graph of a standard function. Analysis of the Order of Transformations for Additional Eample : In order to graph this problem correctly, we need to choose a correct order of transformations. First, decide on the transformations that need to be performed on f (without consideration of correct order). Make a note of whether 3 each transformation has a horizontal or vertical effect on the graph. f 3 f 3 f 3 Reflect in the -ais Shift down Shift right 3 (Vertical Effect) (Vertical Effect) (Horizontal Effect) Remember that any vertically-oriented transformations have no effect on horizontallyoriented transformations, and vice versa. However, the order in which you perform vertically-oriented transformations may make a difference in the graph, and the order in which you perform horizontally-oriented transformations may make a difference in the graph. Notice that the shift to the right is the only transformation that has a horizontal effect on the graph. Since it has no effect on the vertically-oriented transformations, the right shift can be performed at any point in the graphing process. We need to be more careful about the order in which we perform the reflection in the - ais and the downward shift, since they both have a vertical effect on the graph. Remember that you do NOT need to evaluate all the possible combinations in which the above transformations can be performed, but you need to find one which algebraically yields the correct function (and therefore also yields the correct graph). Perform the following transformations algebraically on g to see which one gives the desired function, f 3. (In the choices below, the shift to the right is written first, but we could put that transformation at any point in the process and get the same result.) Choice 1: Shift right 3, reflect in the -ais, shift down g h 3 k 3 f 3 Choice 1 yields the desired function, f 3. MATH 1330 Precalculus 99
47 CHAPTER 1 A Review of Functions Choice : Shift right 3, shift down, reflect in the -ais. g h 3 k 3 f 3 Notice that in Choice, f 3 3 the desired function, f 3., and does NOT yield We can see from the choices above that the reflection in the -ais needs to be performed before the downward shift. Since the right shift is the only horizontally-oriented transformation, it can be performed at any point in the process. Therefore, any of the following combinations of transformations would yield the correct graph: Shift right 3, reflect in the -ais, shift down Reflect in the -ais, shift right 3, shift down Reflect in the -ais, shift down, shift right 3 The solution shown below uses the order of transformations shown in Choice 1. Solution to Additional Eample : 100
48 SECTION 1.3 Transformations of Graphs MATH 1330 Precalculus 101
49 CHAPTER 1 A Review of Functions 10
50 Eercise Set 1.3: Transformations of Graphs Suppose that a student looks at a transformation of y f ( ) and breaks it into the following steps. State the transformation that occurs in each step below. 1. y f ( 3) 4 (a) From: y f ( ) To: y f ( 3) (b) From: y f ( 3) To: y f ( 3) (c) From: y f ( 3) To: y f ( 3) (d) From: y f ( 3) To: y f ( 3) 4 1. y f (1 ) 5 4 First notice that (1 ) is equivalent to ( 1) (a) From: y f ( ) To: y f ( 1) (b) From: y f ( 1) To: y f ( 1) f (1 ) (c) From: y f (1 ) 1 To: y f (1 ) 4 1 (d) From: y f (1 ) 4 1 To: y f (1 ) 4 1 (e) From: y f (1 ) 4 Answer the following. 1 To: y f (1 ) Jack and Jill are graphing the function f ( ). Starting with the graph of y, Jack first reflects the graph in the -ais and then shifts upward two units. Jill, on the other hand, first shifts the graph y upward two units and then reflects in the -ais. Their graphs are shown below. Jack s Graph: y Jill s Graph: y (a) Who is correct, Jack or Jill? (b) Analyze the two methods and eplain the algebraic difference between the two. Use this analysis to justify your answer in part (a). 4. Fred and Wilma are graphing the function f ( ) ( 4). Starting with the graph of y, Fred first reflects the graph in the y-ais and then shifts four units to the left. Wilma, on the other hand, first shifts the graph y four units to the left and then reflects in the y-ais. Their graphs are shown below. Fred s Graph: y Wilma s Graph: (a) Who is correct, Fred or Wilma? (b) Analyze the two methods and eplain the algebraic difference between the two. Use this analysis to justify your answer in part (a). 5. Tony and Maria are graphing the function f ( ). Starting with the graph of y, Tony first shifts the graph two units to the right and then reflects in the y-ais. Maria, on the other hand, first reflects the graph y in the y- ais and then shifts two units to the right. Their graphs are shown below. Tony s Graph: y y (a) Who is correct, Tony or Maria? (b) Analyze the two methods and eplain the algebraic difference between the two. Use this analysis to justify your answer in part (a). Maria s Graph: y MATH 1330 Precalculus 103
51 Eercise Set 1.3: Transformations of Graphs 6. Bart and Lisa are graphing the function f ( ) 3. Starting with the graph of y, Bart first shifts the graph downward three units and then stretches the graph vertically by a factor of. Lisa, on the other hand, first stretches the graph y vertically by a factor of and then shifts the graph downward three units. (a) Who is correct, Bart or Lisa? (b) Analyze the two methods and eplain the algebraic difference between the two. Use this analysis to justify your answer in part (a). Matching. The left-hand column contains equations that represent transformations of f. Match the equations on the left with the description on the right of how to obtain the graph of g from the graph of f. 7. g( ) ( 4) 8. g ( ) 4 9. g ( ) g ( ) ( 4) g( ) g( ) ( ) g( ) g( ) g ( ) g ( ) ( 4) g ( ) ( 3) Bart s Graph: y g ( ) ( 4) Lisa s Graph: y A. Reflect in the -ais. B. Shift left 4 units, then reflect in the y-ais. C. Reflect in the -ais, then shift downward 4 units. D. Shift right 4 units. E. Shift right 3 units, then reflect in the -ais, then shift upward 4 units. F. Shift upward 4 units. G. Reflect in the y-ais. H. Shift left 4 units, then shift upward 3 units. I. Shift left 4 units. J. Shift downward 4 units. K. Stretch vertically by a factor of 4. L. Shrink vertically by a factor of 1 4. Write the equation that results when the following transformations are applied to the given standard function. Then state if any of the resulting functions in (a)-(e) are equivalent. 19. Standard function: 3 y (a) Shift right 7 units, then reflect in the -ais, then stretch vertically by a factor of 5, then shift upward 1 unit. (b) Reflect in the -ais, then shift right 7 units, then stretch vertically by a factor of 5, then shift upward 1 unit. (c) Stretch vertically by a factor of 5, then shift upward 1 unit, then shift right 7 units, then reflect in the -ais. (d) Shift right 7 units, then shift upward 1 unit, then reflect in the -ais, then stretch vertically by a factor of 5. (e) Reflect in the -ais, then shift left 7 units, then stretch vertically by a factor of 5, then shift upward 1 unit. (f) Which, if any, of the resulting functions in (a)- (e) are equivalent? 0. Standard function: y (a) Reflect in the y-ais, then shift left units, then shift downward 4 units, then reflect in the - ais (b) Shift left units, then reflect in the y-ais, then reflect in the -ais, then shift downward 4 units. (c) Reflect in the y-ais, then reflect in the -ais, then shift downward 4 units, then shift right units. (d) Reflect in the -ais, then shift left units, then shift downward 4 units, then reflect in the y- ais. (e) Shift downward 4 units, then shift left units, then reflect in the y-ais, then reflect in the - ais. (f) Which, if any, of the resulting functions in (a)- (e) are equivalent? Continued on the net page 104
52 Eercise Set 1.3: Transformations of Graphs Describe how the graph of g is obtained from the graph of f. (Do not sketch the graph.) 1. f ( ), g ( ) 38. f ( ) f ( ) 3 3. f ( ), 3 g ( ) ( 5) 3. f ( ), g( ) f ( ) ( 5) f ( ) 6 4. f ( ), g ( ) 1 ( 3) f ( ), g ( ) , 3 f g ( ) 4 Describe how the graphs of each of the following functions can be obtained from the graph of y = f(). 7. y f ( ) 1 8. y f ( 7) 9. y f ( ) y f ( 3) y 1 f ( ) 5 3. y 5 f ( ) y f ( 7 ) 34. y f ( 5) 7 4 Sketch the graph of each of the following functions. Do not plot points, but instead apply transformations to the graph of a standard function. 35. f ( ) f ( ) f ( ) f ( ) ( 1) 43. f ( ) 3( 4) 44. f ( ) ( 5) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) ( 4) 1 5. f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) 58. f ( ) MATH 1330 Precalculus 105
53 Eercise Set 1.3: Transformations of Graphs Answer the following. 59. The graphs of f ) 4 ( and g( ) 4 are shown below. Describe how the graph of g was obtained from the graph of f. The graph of y = f() is given below. Sketch the graph of each of the following functions. y f ( ) 4 y g( ) 4 y f 60. The graphs of f ( ) 3 1 and g( ) 1 are shown below. Describe how the graph of g was obtained from the graph of f. 3 y f ( ) 3 1 ( ) 3 y g y f ( ) 64. y f ( ) y f ( ) y f ( 1) y f ( ) Sketch the graphs of the following functions: 68. y f () 61. (a) f ( ) 9 (b) g( ) y f ( ) 6. (a) 1 f ( ) (b) g ( ) y 1 f ( ) 71. y f ( 1) 7. y f ( ) 4 Continued in the net column 106
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