Math 1083 Worksheet 12 Getting Ready for Modeling with Trigonometric Functions Objectives:

1. Identify key features from sinusoidal curves

2. Five key points for the basic sine and cosine graphs

3. Determine the key features of sinusoidal curves from equations

A midline is a horizontal line that divides the graph in half vertically.

Midline Equation for sine and cosine: y = max ??????????+min ??????????

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Amplitude is the distance from the midline to the maximum or minimum.

Amplitude = max ???????????min ??????????

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#1 For each given graph, find the minimum, maximum, amplitude and the midline equation.

a) b)

maximum: ___________ minimum: _________ maximum: ___________ minimum: _________ amplitude: _______ Midline: _______________ amplitude: _______ Midline: _______________

c) d)

maximum: ___________ minimum: _________ maximum: ___________ minimum: _________ amplitude: _______ Midline: _______________ amplitude: _______ Midline: _______________

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Review: the graphs of ?? = sin ?? ?????? ?? = cos ?? over one period, on the interval [0, 2??] #2. Use the graphs of the basic sine and cosine equations to answer the following in terms of “mid” for midline, “max” for maximum, and “min” for minimum.

a) Sine: starting with _________ at ?? = 0

b) Find the height of the five points that correspond to quadrantal points [the first three are

done]

___mid___-> __max_____->__mid____->________->________

c) Cosine: starting with __________ at ?? = 0

d) Find the height of the five points that correspond to quadrantal points

_________-> __________->________->________->________

#3 For each equation below, what is the pattern of the five key points?

a) ?? = 3 sin ?? _________-> __________->________->________->________

b) ?? = ?2 sin ?? _________-> __________->________->________->________

c) ?? = ? cos ?? _________-> __________->________->________->________

d) ?? = 1

2 cos ?? _________-> __________->________->________->________

e) What can you summarize about the starting position (when x = 0) of the sine and cosine

functions? Answer using “min”, “max” or “mid.”

Equation Starting position (when ?? = 0, which is the phase shift in these cases) ?? = ?? sin ??, ?? > 0

?? = ?? sin ??, ?? < 0 ?? = ?? cos ??, ?? > 0

?? = ?? cos ??, ?? > 0

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REVIEW: Transformations of Sine and Cosine Given an equation in the form ??(??) = ?? sin(??(?? ? ??)) + ?? or ??(??) = ?? cos(??(?? ? ??)) + ??

• A is the vertical stretch, and |??| is the amplitude of the function.

• B is the horizontal stretch/compression, and is related to the period, P= 2??