# Modeling periodic behavior

## • Understand the relationship between parameters in a trigonometric function and the shape of the graph (F-IF.C.7e$^\star$).  • Model with trigonometric functions, including fitting them to data (F-TF.B.5$^\star$).

Students have learned the basic shape of the graph of a trigonometric function, and now begin examining graphs of functions with parameters controlling the period, amplitude, and phase shift. They study the effect of varying these parameters and fit trigonometric functions to data.

1 Exploring Sinusoidal Functions

WHAT: Students experiment with the graph of a trigonometric function, using sliders to see how the parameters in the definition of the function affects the graph.

WHY: The purpose of this task is to give students the opportunity to gain familiarity with the effect of different parameters in the definition of a trigonometric function, in particular generalizing the behavior of the amplitude, frequency and period as they observe many examples (MP.8).

## External Resources

1 Moon Safari

#### Description

WHAT: Using data from the US Naval Observatory, students look at the percentage of the moon visible each night, and then use Desmos or another graphing utility (MP.5) to fit a trigonometric function to their data (F-TF.B.5$^\star$).

WHY: This task provides a familiar context where students can connect features of the graph of a trigonometric function with numbers that make sense in the context (MP.4). For instance, the amplitude of the graph will be .5 since the moon varies between 0% and 100% visible, and the period will need to be approximately 30 days to replicate the moon cycle.

2 Siren Song

#### Description

WHAT: Students are presented with a diagram showing sounds waves emanating from a beacon and use it to calculate the frequency and the wavelength (period) of the wave. They then consider the case where the beacon is moving and calculate the effect of its motion on the frequency and wavelength both in front of and behind the moving beacon.

WHY: The purpose of this task is to activate student thinking about the parameters affecting the shape of the graph of a trigonometric function, in particular the period of the graph. In the case of a moving wave, the period is related to the frequency and velocity of the wave, and students explore that relationship in the context of studying the Doppler effect (A-CED.A.2$^\star$). This provides a possible connection with science classes and with future courses in college. This lesson provides a compelling example of a real world phenomenon that can be modeled by a periodic function (MP.4).

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