Lesson 3 of 12
Objective: SWBAT identify the effect on a graph of replacing f(x) +k, k f(x), f(kx) and/of f(x+k) for specific values of k. (both positive and negative)
Set the Stage
I begin with a simple rational function on the board like f(x) = 1/x. I ask my students to predict what the graph will look like without using their calculators. If they don't remember, I suggest they use graph paper and make a table of values, which is sometimes enough to stimulate their memory of parent functions, but which will also give them a good visual if needed. (MP1) When everyone agrees about the general appearance and key features of the function, I challenge them with predicting the appearance of a slightly different function: f(x) = 1/(x-1), again using tables and graph paper. (MP2) I focus on this "by-hand" method for this part of the lesson because many of my students are weak at both creating a table of values from a function and graphing those values. While it's slower going to do it this way, it reinforces necessary skills and students' will have a stronger foundation for understanding my lesson where we use technology for the investigations.
I tell them that today they will have the opportunity to explore how changing a function affects the graph of that function.
Put It Into Action
Students work independently for this part of the lesson to provide me a better idea of which students are struggling and to give them a chance to build their graphing skills and understanding of functions. I explain that they will working with several different functions but that if they're careful they should see patterns in the changes they make. I distribute the handout and review the directions and example problem then check for understanding with fist-to-five. If the majority of the class are still confused I work through an additional example, calling on students to provide the descriptions, then let them begin working. (MP1) While they're working I walk around offering assistance and encouragement as needed. I anticipate that a few students will need help rewriting some of the equations to enter them into their graphing calculator while others who choose to work with tables and graph paper will struggle with values that are not easy to work. For those students I may work through all or part of another example or may remind them that the function notation does NOT imply multiplication. When everyone has completed the handout or when there are about 15 minutes left, I tell them to finish the problem they're on and prepare to participate in a class discussion about their work.
Wrap It Up
To close this lesson I ask my students to pair-share what patterns they found in trying all the different transformations. (MP2) After a moment or two I randomly select students to share what they discussed, acting as scribe to summarize what they say. When everyone has had a chance to speak I ask for volunteers to discuss how changes to "X" and "Y" respectively affect the graph. (MP8) Hopefully someone will observe that adding to or subtracting from "X" moves the graph left or right from it's original position while adding to or subtracting from "Y" moves the graph up or down. Recognizing the impact of multiplying or dividing is more difficult but if nobody comments on these, I ask questions like "what happens if we double x?" or "What happens when we divide y by 3?" The final piece is for each student to write in her/his notes a summary of the effects of each change on a graph of the parent function. My video explains why I choose to let my students figure out these transformation patterns on their own.