Transformations of Functions (Lesson 3.2)
Unit 3 Day 2
CED Topic(s): 1.12
Construct a new function by applying translations, dilations, and reflections to a parent function.
Given an equation or graph of a transformed function, describe the transformations that occurred from the parent function.
Determine the domain and range of a transformed function.
Quick Lesson Plan
Today students work on a Desmos activity to connect analytical and graphical representations of transformations. After having studied six of the primary parent functions in the previous lesson, students are ready to explore how many other functions can be created from the parent functions.
To prepare for today’s lesson, create a class code for the “What’s My Transformation” activity on Desmos. Students will also need colored pencils to trace the transformed parangulas.
Allow students time to explore and test conjectures in this activity. Using Desmos as a tool allows students to make predictions and see the result. Students transition from dragging a “parangola” across the page to translating it using function notation. They observe patterns and are asked to make generalizations about what makes a function move up, down, left, or right. Various screens ask students to summarize their findings. Encourage students to write these summaries on their papers as well.
We have found that having two students share one computer really increases communication and fuels sense making. Slide 13 is challenging, and students benefit from hearing the ideas others have tried.
What do you notice about the equation when the parangula shifts to the right or the left?
What do you notice about the equation when the parangula shifts up or down?
Does the order of the transformations matter?
Why does a negative coefficient flip the parangula?
No margin notes are offered in this activity because we believe the students’ findings are best consolidated in a table like the one offered in the Important Ideas section. This structure note taking allows students to connect symbolic and verbal representations.
Ask students why the notation f(x-2) represents a shift to the right 2 units. Push students beyond an “it’s always the opposite” explanation. Although this concept can be confusing, a concrete example can help. If f(x)=x^2, then the original function has an output of 36 when x=6. However, for the transformed function f(x-2), the output of 36 is attained when x=8. Thus, the same output of 36 occurs two units later than it did on the original function, demonstrating that f(x-2) represents a shift two units to the right. Question 4 on the Check Your Understanding has students reason through this idea with a contextual example. The idea is similar for horizontal dilations. The function f(4x) reaches a given input value four times faster than the parent function f(x). This means that the value of x needed to attain a certain output is one-fourth as big as the x-value needed for f(x) to attain that output. For horizontal translations and dilations, instead of comparing y-values at the same x-value, we are comparing x-values that produce the same y-value.