Lesson 17
Comparing Transformations
Lesson Narrative
In the previous unit students examined how each of the constants \(a, b, h,\) and \(k\) influences the graph of \(g(x) = af(bxh) + k\) for a given function \(f\). This unit has looked in depth at the special case of trigonometric functions. If \(f(x) = \sin(x)\), for example, then \(a\) is the amplitude of \(g\) while \(h\) is a horizontal translation, \(k\) is a vertical translation, and the period of \(g\) is \(\frac{2\pi}{b}\). In this lesson, students examine these parameters and how they influence trigonometric functions and their graphs. This lesson also includes an optional activity using technology for extra practice transforming periodic functions.
The information gap structure encourages students to think carefully about the order in which individual transformations are applied. They also need to think carefully about signs for the translations and about the coefficient of \(x\) in the transformed function to correctly identify how the graph is scaled horizontally. Students have had opportunities to observe the close relationship between transformations of graphs and transformations of functions in the lessons leading up to this one and they now have a chance to apply this knowledge (MP7).
Learning Goals
Teacher Facing
 Determine what information is needed to sketch a transformation of a function and create its equation. Ask questions to elicit that information.
Student Facing
 Let's ask questions to figure out transformations of trigonometric functions.
Required Materials
Required Preparation
If using the optional activity Match the Graph, acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)
Learning Targets
Student Facing
 I can ask questions to figure out how a trigonometric function was transformed.
 I can create an equation of a trigonometric function using information about its graph.
CCSS Standards
Glossary Entries

amplitude
The maximum distance of the values of a periodic function above or below the midline.

midline
The value halfway between the maximum and minimum values of a period function. Also the horizontal line whose \(y\)coordinate is that value.