Tri Math Worksheet

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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 𝑣𝑎𝑙𝑢𝑒+min 𝑣𝑎𝑙𝑢𝑒

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

Amplitude = max 𝑣𝑎𝑙𝑢𝑒−min 𝑣𝑎𝑙𝑢𝑒

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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 𝑦 = sin 𝑥 𝑎𝑛𝑑 𝑦 = cos 𝑥 over one period, on the interval [0, 2𝜋] #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 𝑥 = 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 𝑥 = 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) 𝑦 = 3 sin 𝑥 _________-> __________->________->________->________

b) 𝑦 = −2 sin 𝑥 _________-> __________->________->________->________

c) 𝑦 = − cos 𝑥 _________-> __________->________->________->________

d) 𝑦 = 1

2 cos 𝑥 _________-> __________->________->________->________

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 𝑥 = 0, which is the phase shift in these cases) 𝑦 = 𝑎 sin 𝑥, 𝑎 > 0

𝑦 = 𝑎 sin 𝑥, 𝑎 < 0

𝑦 = 𝑎 cos 𝑥, 𝑎 > 0

𝑦 = 𝑎 cos 𝑥, 𝑎 > 0

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REVIEW: Transformations of Sine and Cosine Given an equation in the form 𝑓(𝑡) = 𝐴 sin(𝐵(𝑡 − 𝐶)) + 𝑘 or 𝑓(𝑡) = 𝐴 cos(𝐵(𝑡 − 𝐶)) + 𝑘

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

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

𝐵

• k is the vertical shift and determines the midline of the function, y =k.

• 𝐶 is the phase (horizontal) shift.

#4 For each of the following equations, find the amplitude, period, phase shift and midline.

a) 𝑦 = 3 sin(8𝑥) + 5 b) 𝑦 = 1

2 cos(3𝑥) − 2

c) 𝑦 = 3 sin ( 𝜋

2 (𝑥 + 1)) − 4 d) 𝑦 = −2 cos (

3𝜋

4 𝑥) + 1

e) 𝑦 = −5 sin(2(𝑥 − 𝜋)) + 3 f) 𝑓(𝑥) = 5 − cos (4𝑥 − 𝜋) 1