Lesson 17

In the previous unit students examined how each of the constants \(a, b, h,\) and \(k\) influences the graph of \(g(x) = af(bx-h) + k\) 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).

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.)