This is the first of two lessons focused on students practicing identifying important features of trigonometric functions when starting from equations or graphs. They also begin to learn how a horizontal scale factor affects the period of these types of functions. While this lesson emphasizes mathematical contexts, students return to working with functions modeling situations in the following lesson with an emphasis toward reasoning about the scale factor affecting the period.
The warmup introduces students to what cosine and sine with different periods can look like. In a later activity, they focus on \(y=\sin(2\theta)\) and use different representations to make sense of why the graph of this function appears horizontally compressed compared to the graph of \(y=\sin(\theta)\).
Students make use of structure when they work on the matching task as they identify important structural properties of graphs and equations (MP7). These properties include midline, amplitude, and horizontal translations and students need to identify these aspects of a function both from the expression defining the function and its graph.
Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
Lesson overview
 15.1 Warmup: Notice and Wonder: Musical Notes (5 minutes)

15.2 Activity: Equations and Graphs (15 minutes)
 Includes "Are you Ready for More?" extension problem
 15.3 Activity: Double the Sine (15 minutes)
 Lesson Synthesis
 15.4 Cooldown: To the left, to the right (5 minutes)
Learning goals:
 Identify the midline, amplitude, and horizontal translation in trigonometric functions presented graphically and with equations.
 Predict how changing the value of \(k\) in \(y=cos(kx)\) changes the graph of the function for whole number values of \(k\).
Learning goals (student facing):
 Let’s compare graphs and equations of trigonometric functions.
Learning targets (student facing):
 I can identify the midline, amplitude, and horizontal translation of a trigonometric function given a graph or equation.
Standards:
 This lesson builds towards the standard: CCSS.HSFIF.C.7.eMS.FIF.7eMO.A2.IF.A.1
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