Lesson 16

This is the second of two lessons focused on students practicing identifying important features of trigonometric functions. In the warm-up and following activity, students continue their thinking about identifying the period of a function from either a graph or an equation.

Periods of real world phenomena are often rational numbers, so trigonometric functions with a horizontal scale factor arise frequently in modeling situations. For example, suppose the function \(f(t) = 4\sin\left(\frac{2\pi t}{3}\right) + 15\) models the vertical position (in feet) of a point at the tip of a windmill blade. Here the input \(t\) is time measured in seconds. Students learn to notice that the input \(\frac{2\pi t}{3}\) changes by a multiple of \(2\pi\) whenever \(t\) changes by a multiple of 3. They understand that this means the period of \(f\) is 3 so the windmill blade completes one revolution every 3 seconds. The amplitude of \(f\) is 4 and the midline is 15: these give the length of the windmill blade and the height of the windmill respectively.

Understanding how to find the period of a function given in equation form requires identifying a repeating pattern whether students think about the unit circle, a table of values, or graphs (MP8).