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2/25/2019
1
Warm up Find the values of the following trig functions:
1. sin ๐ฅ 2. cos๐ฅ
x ๐๐๐ ๐
0
๐
2
๐
3๐
2
2๐
x ๐๐๐ ๐
0
๐
2
๐
3๐
2
2๐
Properties of sine function
Domain: (-โ,โ) Range: [-1,1] y-intercept: 0
x-intercepts: -ฯ, 0, ฯ, etc. Continuity: (-โ,โ) Symmetry: odd
Properties of cosine function
Domain: (-โ,โ) Range: [-1,1] y-intercept: 1
x-intercepts: -ฯ/2, ฯ/2, etc. Continuity: (-โ,โ) 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, ๐ ๐ฅ by examining its general form: ๐ฆ = ๐๐ ๐๐ฅ + ๐ + ๐
The general form of the sinusoidal functions are:
๐ฆ = ๐ sin ๐๐ฅ + ๐ + ๐ and ๐ฆ = ๐ cos ๐๐ฅ + ๐ + ๐
where a, b, c, and d are constants and neither a nor b is 0.
๐ ๐ฅ = sin ๐ฅ ๐ ๐ฅ =1
4sin(๐ฅ)
โ ๐ฅ = 5 sin(๐ฅ) ๐ ๐ฅ = โ2 sin(๐ฅ)
What kind of transformation occurs when ๐ < 1
What kind of transformation occurs when ๐ > 1?
What kind of transformation occurs when ๐ < 0?
Adjusting a: ๐๐ ๐๐ฅ + ๐ + ๐
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Amplitude: half the distance between the maximum and minimum values of thefunction, or half the height ofthe wave
For ๐ฆ = a sin ๐๐ฅ + ๐ + ๐ and ๐ฆ = a cos ๐๐ฅ + ๐ + ๐
amplitude= ๐
๐ ๐ฅ = cos(๐ฅ) ๐ ๐ฅ = sin๐ฅ
3โ ๐ฅ = cos(3๐ฅ)
What kind of transformation occurs when ๐ < 1
What kind of transformation occurs when ๐ > 1?
Bewaring b: ๐๐ ๐๐ฅ + ๐ + ๐
Oscillation: how much a function varies between itโs extreme values
Period: the smallest value, ๐, such ๐ ๐ก + ๐ = ๐ ๐ก for all ๐ก in the domain of ๐ ๐ก ; 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 ๐ฆ = a sin ๐๐ฅ + ๐ + ๐ and ๐ฆ = a cos ๐๐ฅ + ๐ + ๐
period= 2๐
๐
frequency= 1
๐๐๐๐๐๐or
๐
2๐
๐ ๐ฅ = sin(๐ฅ) ๐ ๐ฅ = sin ๐ฅ +๐
2
โ ๐ฅ = sin(3๐ฅ) ๐ ๐ฅ = sin 3๐ฅ โ๐
2
What kind of transformation occurs when c > 0?
What kind of transformation occurs when c < 0?
Considering c: ๐๐ ๐๐ฅ + ๐ + ๐ 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 ๐ฆ = a sin ๐๐ฅ + ๐ + ๐ and ๐ฆ = a cos ๐๐ฅ + ๐ + ๐
phase shift= โ๐
๐
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๐ ๐ฅ = sin(๐ฅ) ๐ ๐ฅ = sin(๐ฅ) + 1 โ ๐ฅ = sin(๐ฅ) โ 1
What kind of transformation occurs when d > 0?
What kind of transformation occurs when d < 0?
Devising d: ๐๐ ๐๐ฅ + ๐ + ๐ 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 ๐ฆ = a sin ๐๐ฅ + ๐ + ๐ and ๐ฆ = a cos ๐๐ฅ + ๐ + ๐
Midline= ๐
Construction from given info Use the given information below to find the
requested function.
1. sine 2. cosineAmplitude: 5 Amplitude: 1
Period: ๐
2Period: 4๐
Phase shift: Left ๐
2Phase shift: Right 2๐
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