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**6.4 Graphs of Sine and Cosine**

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Label your graph paper... 2 1 90º -270º 270º 360º -90º -1 180º -2

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In radians... 2 1 2 -1 -2

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**Graph y = sin sin 0° 0.707 45° 90° 1 135° 0.707 180° 225°**

2 0° 0.707 45° 1 90° 1 -270º -90º 90º 180º 270º 360º 135° 0.707 -1 180° -2 225° -0.707 -1 270° -0.707 315° 360°

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**y = sin x 2 1 -90º 90º -270º 270º -1 -2 Maximum Maximum intercept**

Minimum Minimum -2

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y = sin x Period: the least amount of space (degrees or radians) the function takes to complete one cycle. 2 1 -90º 90º -270º 270º -1 -2 Period: 360°

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**In other words, how high does it go from its axis?**

y = sin x Amplitude: half the distance between the maximum and minimum 2 Amplitude = 1 1 -90º 90º -270º 270º -1 -2 In other words, how high does it go from its axis?

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**Graph y = cos cos 1 0.707 -0.707 -1 -0.707 0.707 1 2 1 -1 -2**

1 2 0.707 1 -0.707 2 -1 -1 -0.707 -2 0.707 2 1

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**y = cos x -2 - 2 2 1 -1 -2 Maximum Maximum intercept intercept**

Minimum Minimum -2

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**y = cos x -2 - 2 Period: 2**

Period: the least amount of space (degrees or radians) the function takes to complete one cycle. 2 1 -2 - 2 -1 -2 Period: 2

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**How high does it go from its axis?**

y = cos x How high does it go from its axis? 2 Amplitude = 1 1 -2 - 2 -1 -2

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**Try it on your calculator!**

y = sin x y = cos x Try it on your calculator! 2 1 -1 -2

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**y= sin and y = cos are the mother functions.**

Changing the equations changes the appearance of the graphs We are going to talk about the AMPLITUDE, TRANSLATIONS, and PERIOD of relative equations

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Mother Function relative function change? y1 = sin x reflection over x-axis y2 = - sin x y1 = sin x y2 = 4 sin x amplitude = 4 amplitude = y2 = sin x y1 = sin x generalization? y = a sin x amplitude = a

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**is the horizontal translation**

Mother Function relative function change? y1 = sin x y2 = sin (x - 45) horizontal translation, 45 degrees to the right. horizontal translation, 60 degrees to the left. y1 = sin x y2 = sin (x + 60) horizontal translation, 30 degrees to the left. y2 = sin (2x + 60) y1 = sin x horizontal translation, 90 degrees to the right. y1 = sin x y2 = sin (3x - 270) generalization? y = sin (bx - c) to the right is the horizontal translation y = sin (bx – (- c)) to the left

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**‘d’ is the vertical translation**

Mother Function relative function change? y1 = cos x y2 = 2 + cos x vertical translation, units up. vertical translation, units down. y1 = cos x y2 = -3 + cos x generalization? y = d + cos x ‘d’ is the vertical translation when d is positive, the graph moves up. when d is negative, the graph moves down.

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Mother Function relative function change? y1 = sin x y2 = sin 2x Period = 180 or Period = 720 or y1 = sin x y2 = sin x generalization? Period = y = sin bx or

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**Summary: y = d + a sin (bx - c) y = d + a cos (bx - c)**

a is the amplitude period = or is the horizontal translation d is the vertical translation

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**Analyze the graph of amplitude = period = horizontal translation:**

vertical translation: none

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**Analyze the graph of 3 (to the left) amplitude = period =**

horizontal translation: vertical translation: none

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**Analyze the graph of 3 amplitude = period = horizontal translation:**

none vertical translation: Up 2

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**Graph and Analyze y = -2 + 3 cos (2x - 90°) 3 x y high 1 amplitude =**

45° 3 amplitude = 90° -2 mid period = = 180° 135° low -5 horizontal translation: 180° -2 mid (to the right) 225° 1 high vertical translation: down 2 1) horiz. tells you where to start 3) divide period by 4 to find increments = 45 2) add the period to find out where to finish table goes in increments of 45 4) plot points and graph = 225

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**You must know how to analyze the equation before you can graph it.**

The most important thing to remember about graphing is determining the starting point and the stopping point on the t-table.

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**Ex #6c Graph y = 1 + 3 sin (2 + ) = 3 up 1 x y mid 1 amplitude =**

high = 4 period = 1 mid horizontal translation: -2 low up 1 vertical translation: mid 1 1) horiz. tells you where to start On Calculator, go to table setup and change independent to ask. 3) divide period by 4 to find increments table goes in increments of 2) add the period to find out where to finish 4) plot points and graph

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Label each of the following graphs with the appropriate function. Calculator should be set to radians. Window Xscl should be set to pi. The amplitude equals:

Label each of the following graphs with the appropriate function. Calculator should be set to radians. Window Xscl should be set to pi. The amplitude equals:

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