What Is A Function



Activity 6.5 Sinusoidal Functions

Overview:

This activity has exercises that review all characteristics of the graphs of the sine and cosine functions and can be given as an in-class of out-of-class assignment. It includes a chart in which students are to list the key characteristics of the graphs of different forms of the sine and cosine functions and it has 11 exercises in which students are to sketch the graph of sine or cosine functions. Question 12 explores harmonic motion of tides and asks to students to graph a specific situation and write an equation that models the behavior. The last question asks students to write equivalent equations to given functions using both sine and cosine.

Estimated Time Required:

This activity should take approximately 30 – 45 minutes.

Technology: None

Prerequisite Concepts:

• amplitude

• period

• horizontal and vertical shifts

• horizontal and vertical stretches and compressions

Discussion:

The main point of this activity is to help students understand how the algebraic changes in the sine and cosine functions are reflected in changes in the appearance of the sine and cosine graphs.

Point out that many periodic phenomenon that we wish to model have periods different from 2(. For example, the tide might have a period of 12 hours or a pendulum might have a period of 3 seconds. Therefore, it is necessary to have graphs that have the shape of a cosine or sine function but have different periods.

Students often have difficulty deciding which function – sine or cosine – to use when they attempt to find formulas that describe graphs. Emphasize that either is acceptable, and that the only area where this will cause a difference is in the horizontal shift. Students may have difficulty seeing this. Showing them that sine and cosine are really just horizontal shifts of each other may be helpful here.

Activity 6.5 Sinusoidal Functions

• Describe the key characteristics of the graphs of the following functions by listing the period, phase shift, amplitude, and the form of all zeros.

|Function |Key Characteristics |

|[pic] | |

|[pic] | |

|[pic] | |

|[pic] | |

|[pic] | |

|[pic] | |

|[pic] | |

|[pic] | |

Sketch the following functions, and describe how each one is related to the basic sine or cosine graph:

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

10. [pic] 11. [pic]

12. Suppose the tide rises 15 feet above and below mean sea level. The tide pattern repeats every 12 hours. If the tide is at its lowest point at 10:00 AM, find a formula giving the height of the tide (relative to mean sea level) as a function of the number of hours since 10:00 AM. Also, graph one period of the function that you have found. Label your graph clearly.

To do this, find the following:

• period

• amplitude

• any shifts or reflections

13. The graphs of the sine and cosine functions can be thought of as horizontal shifts of each other.

a.) Find h such that the graph of [pic] will be the same graph as that of [pic].

b.) Find h such that the graph of [pic] will be the same graph as that of [pic].

c.) Find h such that the graph of [pic] will be the same graph as that of [pic].

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