# Lesson 16

Features of Trigonometric Graphs (Part 2)

### Lesson Narrative

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).

### Learning Goals

Teacher Facing

• Identify the period of a trigonometric function from its graph or equation.
• Interpret a trigonometric function modeling a situation.

### Student Facing

• Let's explore a trigonometric function modeling a situation.

### Student Facing

• I can find the period of a trigonometric function using an equation or graph.

Building On