Download Translations and Mapping Notation

Name______________________________________________________________________Date_____________Period_______A#_B-­‐14_ Translations and Mapping Notation 1. Using your graphing calculator, graph 𝑦! = 𝑥 ! . 2. Graph 𝑦! = 𝑥 ! . Now graph 𝑦! = (𝑥 + 4)! . What is !
the effect of replacing 𝑥 with (𝑥 + 4)? Sketch the Now graph 𝑦! = (𝑥 − 3) . What is the effect of replacing 𝑥 with (𝑥 − 3)? Sketch the graphs with graphs with arrows to show the translation. arrows to show the translation. 3. Repeat 1 and 2 above, but use the square root function. That is, graph 𝑦! = 𝑥 and compare it to 𝑦! = 𝑥 − 3. Then graph 𝑦! = 𝑥 and compare it to 𝑦! = 𝑥 + 4. Write a generalization of what happens to the graph of 𝑦 = 𝑓(𝑥) when 𝑥 is replaced with (𝑥 − ℎ) to get 𝑦 = 𝑓(𝑥 − ℎ). (Assume that ℎ > 0.) A sketch may be helpful. !
5. Graph 𝑦! = 𝑥 . Now graph 𝑦! = 𝑥 − 4. This is 4. Using your graphing calculator, graph 𝑦! = !. equivalent to 𝑦! + 4 = 𝑥 . What is the effect of !
Now graph 𝑦! = ! + 3. This is equivalent to replacing 𝑦 with (𝑦 + 4)? Sketch the graphs with !
arrows to show the translation. 𝑦! − 3 = !. What is the effect of replacing y with (y – 3)? Sketch the graphs with arrows to show the translation. 6. Write a generalization of what happens to the graph of 𝑦 = 𝑓(𝑥) when 𝑦 is replaced with (𝑦 − 𝑘) to get 𝑦 − 𝑘 = 𝑓(𝑥). (Assume that 𝑘 > 0.) !
8. Write a generalization of what happens to the 7. Using your graphing calculator, graph 𝑦! = ! ! . graph of 𝑦 = 𝑓(𝑥) when 𝑥 is replaced with !
Now graph 𝑦! = (!!!)! + 2. This is equivalent to (𝑥 − ℎ) and 𝑦 is replaced with (𝑦 − 𝑘) to get !
𝑦 − 𝑘 = 𝑓(𝑥 − ℎ). (Assume that ℎ > 0 and 𝑘 > 0.) 𝑦! − 2 = (!!!)! . What is the effect of replacing 𝑥 with (𝑥 − 5) and 𝑦 with (𝑦 – 2)? Sketch the graphs with arrows to show the translation. Adapted from the work of Nils Ahbel. www.ahbel.com Your answers to #1 and #4 should have been something like, “The graph moved 3 units to the right” and “The graph moved 3 units up”. Mapping notation is convenient to communicate translations. We say… In mapping notation this looks like… “The graph is translated 3 units to the right.” 𝑇: (𝑥, 𝑦) → (𝑥 + 3, 𝑦) “The graph is translated 3 units up.” 𝑇: (𝑥, 𝑦) → (𝑥, 𝑦 + 3) 𝑇 : (𝑥, 𝑦) → (𝑥 + 3, 𝑦) “The transformation such that any point (𝑥, 𝑦) maps to (𝑥 + 3, 𝑦).” 9. Write a rule in mapping notation for the transformation that maps 𝑓(𝑥) = 𝑙𝑜𝑔(𝑥) to 𝑓(𝑥) = 𝑙𝑜𝑔(𝑥 + 4). Note: you do not need to know anything about the graph of 𝑓(𝑥) = 𝑙𝑜𝑔(𝑥) in order to do this problem. 10. Write a rule in mapping notation for the transformation that maps 𝑓 𝑥 = 2! to 𝑓 𝑥 = 2!!! . Note: you do not need to know anything about the graph of 𝑓 𝑥 = 2! in order to do this problem. 11. The graph of the function below is the graph of 𝑓(𝑥) = |𝑥| after the transformation 12. The graph of the function below is the graph of 𝑓(𝑥) = 𝑥 ! after the transformation 𝑇: (𝑥, 𝑦) → (𝑥 − 3, 𝑦 + 1). 𝑇: (𝑥, 𝑦) → (𝑥 + 1, 𝑦 + 2). What is the equation of the graph below? Note the graph passes through (1,5). Test to see if your equation includes (1,5). If not, you made a mistake. What is the equation of the graph below? Test your answer using the method in #11. Adapted from the work of Nils Ahbel. www.ahbel.com