# UTPA STEM/CBI Courses/Calculus/Modeling Periodic Behavior

Course Title: Precalculus

Lecture Topic: Modeling Periodic Behavior

Instructor: Virgil U. Pierce

Institution: University of Texas -- Pan American

## Backwards Design[edit | edit source]

**Course Objectives**

**Primary Objectives**- After this module students will be able to:- Find sinusoidal functions which model periodic data collected from an experiment.
- Determine a graph from a sinusoidal function.
- Understand some basic concepts of 'Best Fit' of a function to slightly a-periodic data.

**Sub Objectives**- The objectives will require that students be able to:- Graph sine and cosine functions.
- Identify the amplitude, average value, period and phase shift of periodic data or a periodic function.

**Difficulties**-- Particular care needs to be taken, and this module is designed particularly to address, the determining of a phase shift of a standard sine or cosine function to match periodic data.
- Students are often confused about the relationships between the
**wave number**,**period**and**frequency**

**Real-World Contexts**- There are many ways that students can use this material in the real-world, such as:- Determining functions for making predictions from data collected in: biology, medicine, physics, astronomy, sociology, mechanics and other periodic or almost periodic behaviors.

**Model of Knowledge**

**Concept Map***Amplitude*and*average value*of a sinusoidal function.*Maximum*and*minimum*of a sinusoidal function.*Period*,*frequency*and*wave number*of a sinusoidal function.*Phase shift*of a sinusoidal function- More generally transformations of graphs such as
*translation*,*dilation*and*reflection*.

**Content Priorities****Enduring Understanding**- Determine the parameters of a sinusoidal function.
- Use simple algebra to solve for the wave number from the period or frequency and vice versa.

**Important to Do and Know**- Draw a scatter plot, possibly using technology such as a
*Speadsheet*or*Mathematica*type application.

- Draw a scatter plot, possibly using technology such as a
**Worth Being Familiar with**- Translation of graphs of equations
- Dilation of graphs of equations
- Reflection of graphs of equations

**Assessment of Learning**

**Formative Assessment**- In Class (groups)
- Plotting Sine and Cosine Functions and manipulating the parameters (experiments) -- best done with the help of some technology, possibly in a computer lab with
*Mathematica*. - Drawing a scatter plot of the data.
- Determining the parameters of our plotted data.
- Group agreement on the correct function to match the parameters.
- Use of a computer program such as
*Mathematica*to manipulate the phase parameter. - Determining what is meant by best fit for an example with a-periodic data.

- Plotting Sine and Cosine Functions and manipulating the parameters (experiments) -- best done with the help of some technology, possibly in a computer lab with
- In Class (clickers)
- Matching graphs to functions and vice versa.

- Homework (individual)
- Matching graphs to functions and vice versa.
- Repeating the procedure with different experimental data.

- In Class (groups)
**Summative Assessment**- Test or Quiz question on matching a sine function to given data.
- It could be just matching to a given Maximum and Minimum.
- It could be a more generally small version of the
*challenge problem*.

- Test or Quiz question on matching a sine function to given data.

## Legacy Cycle[edit | edit source]

**OBJECTIVE**

By the next class period, students will be able to:

- Construct a periodic function matching data,
- Identify important parameters (period, amplitude) in the data,
- Discuss the closeness of a periodic model of data to real data.

The objectives will require that students be able to:

- Identify an appropriate variable and time scale,
- Find the amplitude of periodic data,
- Find the period of periodic data,
- Find the vertical shift of the data,
- Find the phase shift to match a function to the data.

**THE CHALLENGE**

Predator-Prey interactions often exhibit a periodic behavior in the populations of the predators and prey. You are a hungry fox who subsists largely on rabbits. The following data was collected in 2010-2011 from a field at the far edge of your range. When is the best time of year to visit the field?

The following data was collected in 2010-2011:

Month | January '10 | February '10 | March '10 | April '10 | May '10 | June '10 | July '10 | August '10 | September '10 | October '10 | November '10 | December '10 |

Population of Rabbits | 11 | 11 | 14 | 21 | 28 | 33 | 38 | 40 | 37 | 28 | 21 | 15 |

Month | January '11 | February '11 | March '11 | April '11 | May '11 | June '11 | July '11 | August '11 | September '11 | October '11 | November '11 | December '11 |

Population of Rabbits | 13 | 9 | 12 | 17 | 24 | 32 | 35 | 39 | 40 | 32 | 20 | 17 |

**GENERATE IDEAS**

- Students should identify the data as likely being periodic, best method is to sketch a graph of the data points.
- Students should then identify the key parameters needed to specify a periodic function.
- A sub lesson should be spent on asking students to graph various trigonometric functions to identify where the parameters are in an expression.

- Students should get used to constructing a graph of the data in order to easily identify the key elements.

**MULTIPLE PERSPECTIVES**

- The most challenging part of the problem is incorporating the phase shift in the data. Students should discuss how this can be done.
- There are many different solutions.
- Students should discuss why.
- The periodic models will not fit the data perfectly. Students should discuss why.

**RESEARCH & REVISE**

Students should now:

- Identify the
**Maximum**and**Minimum**of the data. - Identify an
**amplitude**,**vertical shift**and**period**of the data, - Identify a candidate function that incorporates all of the parameters except for the phase shift.
- Identify the appropriate phase shift.
- Test their answer by graphing it.

Interested students could be asked to study why predator-prey interactions can lead to periodic behavior.

**TEST YOUR METTLE** and **Go Public**

Modify the data above. In practice each group of students should get different data, the groups can then exchange their data and compare their answers.

## Pre-Lesson Quiz[edit | edit source]

- Sketch a graph of f(t) = sin (3t)
- Sketch a graph of f(t) = sin (6t)
- Sketch a graph of f(t) = sin (2πt/2)
- Sketch a graph of f(t) = 2 sin (t)
- Sketch a graph of f(t) = 1/2 sin (t)
- Sketch a graph of f(t) = sin (t) +5
- Sketch a graph of f(t) = sin (t-π/4)
- Sketch a graph of f(t) = sin (2π/2 (t-1/4))

## Test Your Mettle Quiz[edit | edit source]

- Find a trigonometric function with amplitude 20, maximum value 50, and period 100.
- Find a trigonometric function with amplitude 20, maximum value 50, and period 100; where the maximum occurs at t = 50.
- Find a trigonometric function with amplitude 20, maximum value 50, and period 100; where the maximum occurs at t = 15.
- Give a trigonometric function which models the population of rabbits in the field,
- Identify the best time of year to visit the field as the fox.