Solutions for Transformations of Functions

Transcription

1 Solutions for Transformations of Functions I. Souldatos February 20, 209 Answers Problem... Let f(x) = (x + 3) x (x ). Match the following compositions with the functions below. A. f(x + 2) B. f(x 2) C. f(x) + 2 D. f(x) 2 E. 2f(x) F. 2 f(x) G. f(2x) H. f( 2 x) I. f( x) J. f(x) Function Matching Composition from A-J. (x + 3) x (x ) 2 D 2. ((x 2) + 3) (x 2) ((x 2) ) B 3. 2 (x + 3) x (x ) E 4. (2x + 3) 2x (2x ) G 5. (x + 3) x (x ) + 2 C 6. ((x + 2) + 3) (x + 2) ((x + 2) ) A 7. (x + 3) x (x ) J 8. ( x + 3) ( x) ( x ) I 9. 2 (x + 3) x (x ) F 0. ( x 2 + 3) x 2 ( x 2 ) H Grade yourself: One point for each correct answer. Subtotal: /0

2 ANSWERS.2. Now draw f(x) and each of the functions from ()-(0) on the same graph and decide how the graph of f(x) is transformed each time. The possible transformations are i. Move right ii. Move left iii. Move up iv. Move down v. Stretch horizontally vi. Shrink horizontally vii. Stretch vertically viii. Shrink vertically ix. Flip over the x-axis x. Flip over the y-axis The graph of is the graph of f(x)... (x + 3) x (x ) 2 moved down by 2 ((x 2) + 3) (x 2) ((x 2) ) moved right by 2 2 (x + 3) x (x ) stretched vertically by a factor of 2 (2x + 3) 2x (2x ) shrunk horizontally by a factor of 2 (x + 3) x (x ) + 2 moved up by 2 ((x + 2) + 3) (x + 2) ((x + 2) ) moved left by 2 (x + 3) x (x ) flipped about the x-axis ( x + 3) ( x) ( x ) flipped about the y-axis 2 (x + 3) x (x ) shrunk vertically by a factor of 2 ( x 2 + 3) x 2 ( x 2 ) stretched horizontally by a factor of 2 Grade yourself: One point for each correct answer. Subtotal: /0.3. Now put your answers in the previous two questions together Composition Corresponding transformation f(x + 2) Move left by 2 f(x 2) Move right by 2 f(x) + 2 Move up by 2 f(x) 2 Move down by 2 2f(x) Stretch vertically by a factor of 2 2 f(x) f(2x) Shrink vertically by a factor of 2 Shrink horizontally by a factor of 2 f( 2x) Stretch horizontally by a factor of 2 f( x) Flip over the y-axis f(x) Flip over the x-axis Grade yourself: One point for each correct answer. Subtotal: /0 Page 2 of 5

3 ANSWERS To help with the next problem, make sure you got the last table right. Composition Corresponding transformation f(x + 2) Move left by 2 f(x 2) Move right by 2 f(x) + 2 Move up by 2 f(x) 2 Move down by 2 2f(x) Stretch vertically by a factor of 2 2 f(x) f(2x) Shrink vertically by a factor of 2 Shrink horizontally by a factor of 2 f( 2x) Stretch horizontally by a factor of 2 f( x) Flip over the y-axis f(x) Flip over the x-axis Notice: Anything that affects x results in a horizontal transformation, while anything that affects f(x) results in a vertical transformation. While the transformations of f(x) behave as expected, e.g. multiplying f(x) by 2 stretches the graph vertically by a factor of 2, the transformation of x behave the exact opposite, e.g. multiplying x by 2 shrinks the graph horizontally. Problem The graph of f(x) is given below. Match the following formulas with the graphs below.. f(x) f(x) 3. f(x + 3) 4. f(x ) 5. f(x) 6. f(x 3) + 2 Page 3 of 5

4 ANSWERS Graph Formula Graph Formula Grade yourself: One point for each correct answer. Subtotal: / What transformations are needed to transform f(x) to 2f(x + 3) 5? First move f(x) to the left by 3. Then stretch the graph vertically by a factor of 2. Then move the graph down by 5. Grade yourself: One point for each correct answer. Subtotal: / The graph of g(x) is the graph of f(x) (i) stretched vertically by a factor of 2 and (ii) shrunk horizontally by a factor of 3. Find a formula for g(x) in terms of the function f(x). g(x)= 2f(3x). Grade yourself: Two points for the right answer. One point for partially correct answers. Subtotal: /2 Page 4 of 5

5 ANSWERS 2.4. The function f(x) has domain [ 2, 6] and range [ 8, 6]. (a) If a horizontal shrink by a factor of 3 is applied to f(x), what is the domain of the new function? What is the range? Domain= [ 2 3, 2] Range= [ 8, 6] The domain became 3 times smaller, while the range was not changed. (b) If a vertical stretch by a factor of 4 is applied to f(x), what is the domain of the new function? What is the range? Domain= [ 2, 6] Range= [ 32, 24] The range now became 4 times bigger, while the domain was not changed. (c) Use your answers in the previous two questions to find the domain and the range of 4f(3x). Notice that 4f(3x) implies a horizontal shrink by a factor of 3 and a vertical stretch by a factor of 4. These are the last two transformations together. Since the first transformation affects only the domain and the second transformation affects only the range, 4f(3x) has domain= [ 2 3, 2] and range= [ 32, 24]. Grade yourself: One point for each correct answer. Subtotal: /6 Page 5 of 5