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Warm up Find the values of the following trig functions:

1. sin 𝑥 2. cos𝑥

x 𝒔𝒊𝒏 𝒙

0

𝜋

2

𝜋

3𝜋

2

2𝜋

x 𝒄𝒐𝒔 𝒙

0

𝜋

2

𝜋

3𝜋

2

2𝜋

Properties of sine function

Domain: (-∞,∞) Range: [-1,1] y-intercept: 0

x-intercepts: -π, 0, π, etc. Continuity: (-∞,∞) Symmetry: odd

Properties of cosine function

Domain: (-∞,∞) Range: [-1,1] y-intercept: 1

x-intercepts: -π/2, π/2, etc. Continuity: (-∞,∞) Symmetry: even

Sinusoid: any transformation of a sine function

Is there a relationship between the graph of the sine and cosine functions?

Recall that in Unit 1, we transformed a function, 𝑓 𝑥 by examining its general form: 𝑦 = 𝑎𝑓 𝑏𝑥 + 𝑐 + 𝑑

The general form of the sinusoidal functions are:

𝑦 = 𝑎 sin 𝑏𝑥 + 𝑐 + 𝑑 and 𝑦 = 𝑎 cos 𝑏𝑥 + 𝑐 + 𝑑

where a, b, c, and d are constants and neither a nor b is 0.

𝑓 𝑥 = sin 𝑥 𝑔 𝑥 =1

4sin(𝑥)

ℎ 𝑥 = 5 sin(𝑥) 𝑘 𝑥 = −2 sin(𝑥)

What kind of transformation occurs when 𝑎 < 1

What kind of transformation occurs when 𝑎 > 1?

What kind of transformation occurs when 𝑎 < 0?

Adjusting a: 𝒂𝑓 𝑏𝑥 + 𝑐 + 𝑑

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Amplitude: half the distance between the maximum and minimum values of thefunction, or half the height ofthe wave

For 𝑦 = a sin 𝑏𝑥 + 𝑐 + 𝑑 and 𝑦 = a cos 𝑏𝑥 + 𝑐 + 𝑑

amplitude= 𝑎

𝑓 𝑥 = cos(𝑥) 𝑔 𝑥 = sin𝑥

3ℎ 𝑥 = cos(3𝑥)

What kind of transformation occurs when 𝑏 < 1

What kind of transformation occurs when 𝑏 > 1?

Bewaring b: 𝑎𝑓 𝒃𝑥 + 𝑐 + 𝑑

Oscillation: how much a function varies between it’s extreme values

Period: the smallest value, 𝑐, such 𝑓 𝑡 + 𝑐 = 𝑓 𝑡 for all 𝑡 in the domain of 𝑓 𝑡 ; one cycle.

The distance between sets of repeating points on the graph of the function

Frequency: the number of cycles the function completes in a one unit interval.

The frequency is the reciprocal of the period.

For 𝑦 = a sin 𝑏𝑥 + 𝑐 + 𝑑 and 𝑦 = a cos 𝑏𝑥 + 𝑐 + 𝑑

period= 2𝜋

𝑏

frequency= 1

𝑝𝑒𝑟𝑖𝑜𝑑or

𝑏

2𝜋

𝑓 𝑥 = sin(𝑥) 𝑔 𝑥 = sin 𝑥 +𝜋

2

ℎ 𝑥 = sin(3𝑥) 𝑘 𝑥 = sin 3𝑥 −𝜋

2

What kind of transformation occurs when c > 0?

What kind of transformation occurs when c < 0?

Considering c: 𝑎𝑓 𝑏𝑥 + 𝒄 + 𝑑 Phase: the position of a wave relative to some reference

point. A horizontal translation of a sinusoidal function results in a phase shift.

Phase shift: the difference between the horizontal position of the function and that of an otherwise similar sinusoidal function

For 𝑦 = a sin 𝑏𝑥 + 𝑐 + 𝑑 and 𝑦 = a cos 𝑏𝑥 + 𝑐 + 𝑑

phase shift= −𝑐

𝑏

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𝑓 𝑥 = sin(𝑥) 𝑔 𝑥 = sin(𝑥) + 1 ℎ 𝑥 = sin(𝑥) − 1

What kind of transformation occurs when d > 0?

What kind of transformation occurs when d < 0?

Devising d: 𝑎𝑓 𝑏𝑥 + 𝑐 + 𝒅 Vertical Shift: the vertical translation

Midline: horizontal axis that is used as the reference line about which the graph of periodic function oscillates

Equal to the average of the maximum and minimum values of the function

For 𝑦 = a sin 𝑏𝑥 + 𝑐 + 𝑑 and 𝑦 = a cos 𝑏𝑥 + 𝑐 + 𝑑

Midline= 𝑑

Construction from given info Use the given information below to find the

requested function.

1. sine 2. cosineAmplitude: 5 Amplitude: 1

Period: 𝜋

2Period: 4𝜋

Phase shift: Left 𝜋

2Phase shift: Right 2𝜋

Vertical shift: Up 2 Vertical Shift: Down 3

Real world example Musical notes are classified by frequency. In the equal

tempered scale, middle C has a frequency of 262 Hz. In physics, frequency is measured in hertz or oscillations per second. Use this information to write an equation for a sine function that can be used to model the initial behavior of the sound wave associated with middle C having an amplitude of 0.2.

Construction from Graphs Using the following graphs:

Determine if sine or cosine would be bestfor the graph.

Determine the amplitude, period, phase shift, and midline.

Write an equation for each graph.

Construction from Graphs Using the following graphs:

Determine if sine or cosine would be bestfor the graph.

Determine the amplitude, period, phase shift, and midline.

Write an equation for each graph.

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Fun Activities to do at Home (HW)

Unit 4.4 Worksheet #8-14 Even

pp264-265 #2, 6, 10-18 Even