Performing transformations #2 - Questions and Answers

Question 1 Lily wants to define a transformation (or series of transformations) using only rotations, reflections, or translations that takes Figure A to Figure B. Which statement about the transformation that Lily wants to define is true? A. It can be defined with two reflections. B. It can be defined with one rotation and one translation. C. It cannot be defined because Figure A and Figure B are not congruent. D. It cannot be defined because the longest side of Figure B is on the bottom. Answer: Step 1: Observe that the quadrilateral in Figure A lacks any parallel sides. The Figure B is a quadrilateral with a pair of opposite sides parallel. That is, Figure B is a trapezium. The Figure A and Figure B are not congruent since A is a quadrilateral and B is a trapezium. Therefore, Figure A cannot be transformed to Figure B by using only rotations, reflections, or translations. Step 2: The correct option is C. Because Figure A and Figure B are not identical, it cannot be defined. Question 2 Determine the matrix representation of each of the following composite transformations. A pitch of 9 0 ∘ 90 ∘ followed by a yaw of 9 0 ∘ 90 ∘ . Answer: If L is pitch transformation with the angle of theta, the matrix representation of L is given by: 𝑝 = [ cos ⁡ 𝜃 sin ⁡ 𝜃 0 0 − sin ⁡ 𝜃 cos ⁡ 𝜃 0 0 0 0 1 0 0 0 0 1 ] p= ​ cosθ −sinθ 0 0 ​ sinθ cosθ 0 0 ​ 0 0 1 0 ​ 0 0 0 1 ​ ​ If L is the pitch transformation with the angle of rotation phi, the matrix representation of L is given by: 𝑝 𝑦 = [ cos ⁡ 𝜙 0 sin ⁡ 𝜙 0 0 1 0 0 − sin ⁡ 𝜙 0 cos ⁡ 𝜙 0 0 0 0 1 ] . py= ​ cosϕ 0 −sinϕ 0 ​ 0 1 0 0 ​ sinϕ 0 cosϕ 0 ​ 0 0 0 1 ​ ​ . Substitute 9 0 ∘ 90 ∘ : 𝑝 = [ 0 1 0 0 − 1 0 0 0 0 0 1 0 0 0 0 1 ] , 𝑝 𝑦 = [ 0 0 1 0 0 1 0 0 − 1 0 0 0 0 0 0 1 ] . p= ​ 0 −1 0 0 ​ 1 0 0 0 ​ 0 0 1 0 ​ 0 0 0 1 ​ ​ ,py= ​ 0 0 −1 0 ​ 0 1 0 0 ​ 1 0 0 0 ​ 0 0 0 1 ​ ​ . Combine the matrices: [ 0 1 0 0 − 1 0 0 0 0 0 1 0 0 0 0 1 ] [ 0 0 1 0 0 1 0 0 − 1 0 0 0 0 0 0 1 ] = [ 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 ] . ​ 0 −1 0 0 ​ 1 0 0 0 ​ 0 0 1 0 ​ 0 0 0 1 ​ ​ ​ 0 0 −1 0 ​ 0 1 0 0 ​ 1 0 0 0 ​ 0 0 0 1 ​ ​ = ​ 0 0 1 0 ​ 1 0 0 0 ​ 0 1 0 0 ​ 0 0 0 1 ​ ​ . Question 3 Explain how you could graph each function by applying transformations. (a) 𝑦 = log ⁡ ( 𝑥 − 2 ) + 7 y=log(x−2)+7 (b) 𝑦 = − 3 log ⁡ 𝑥 y=−3logx (c) 𝑦 = log ⁡ ( − 3 𝑥 ) − 5 y=log(−3x)−5 Answer: (a) For 𝑦 = log ⁡ ( 𝑥 − 2 ) + 7 y=log(x−2)+7: Start with the graph 𝑦 = log ⁡ ( 𝑥 ) y=log(x). Shift the graph horizontally right 2 units: 𝑦 = log ⁡ ( 𝑥 − 2 ) y=log(x−2). Shift the graph vertically up 7 units: 𝑦 = log ⁡ ( 𝑥 − 2 ) + 7 y=log(x−2)+7. (b) For 𝑦 = − 3 log ⁡ 𝑥 y=−3logx: Start with the graph 𝑦 = log ⁡ ( 𝑥 ) y=log(x). Reflect 𝑦 = log ⁡ ( 𝑥 ) y=log(x) about the x-axis: 𝑦 = − log ⁡ ( 𝑥 ) y=−log(x). Stretch vertically by a factor of 3: 𝑦 = − 3 log ⁡ ( 𝑥 ) y=−3log(x). (c) For 𝑦 = log ⁡ ( − 3 𝑥 ) − 5 y=log(−3x)−5: Start with the graph 𝑦 = log ⁡ ( 𝑥 ) y=log(x). Reflect 𝑦 = log ⁡ ( 𝑥 ) y=log(x) about the y-axis: 𝑦 = log ⁡ ( − 𝑥 ) y=log(−x). Dilate horizontally by a factor of 3: 𝑦 = log ⁡ ( − 3 𝑥 ) y=log(−3x). Shift vertically down by 5 units: 𝑦 = log ⁡ ( − 3 𝑥 ) − 5 y=log(−3x)−5. Question 4 List a sequence of transformations, in words, to convert the graph of 𝑓 ( 𝑥 ) = 𝑥 2 f(x)=x 2 into the graph of 𝑔 ( 𝑥 ) = − 6 ( 𝑥 − 2 ) 2 + 5 g(x)=−6(x−2) 2 +5. Answer: Given: 𝑓 ( 𝑥 ) = 𝑥 2 f(x)=x 2 , convert to 𝑔 ( 𝑥 ) = − 6 ( 𝑥 − 2 ) 2 + 5 g(x)=−6(x−2) 2 +5. Stretch the graph vertically by a factor of 6: 𝑦 = 6 𝑥 2 y=6x 2 . Reflect the graph vertically (about the x-axis): 𝑦 = − 6 𝑥 2 y=−6x 2 . Shift the graph horizontally to the right by 2 units: 𝑦 = − 6 ( 𝑥 − 2 ) 2 y=−6(x−2) 2 . Shift the graph vertically upward by 5 units: 𝑦 = − 6 ( 𝑥 − 2 ) 2 + 5 y=−6(x−2) 2 +5. Thus, the transformations are: Vertical stretch, vertical reflection, horizontal shift (right), and vertical shift (up).