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6.4 Graphs of Sine and Cosine

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90º 270º 180º º -270º -90º Label your graph paper...

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22 In radians...

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Graph y = sin 90º-90º 270º-270º -2 sin 0°0° 45° 90° 135° 180° 225° 270° 315° 360° º360º 2 1

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y = sin x 90º-90º 270º-270º Maximum Minimum intercept

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

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

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Graph y = cos cos 22 22

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y = cos x 2 -- -2 Maximum Minimum intercept

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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.

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

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y = cos xy = sin x Try it on your calculator!

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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 Functionrelative functionchange? y 1 = sin x y 2 = - sin x reflection over x-axis y 1 = sin x y 2 = 4 sin x y 2 = sin x amplitude = 4 amplitude = generalization? y = a sin x amplitude = a

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Mother Functionrelative functionchange? y 1 = sin x y 2 = sin (x - 45) y 2 = sin (x + 60) horizontal translation, 45 degrees to the right. horizontal translation, 60 degrees to the left. y 1 = sin x generalization? y = sin (bx - c) y = sin (bx – (- c)) is the horizontal translation to the right to the left y 2 = sin (2x + 60) y 1 = sin x horizontal translation, 30 degrees to the left. y 2 = sin (3x - 270)y 1 = sin x horizontal translation, 90 degrees to the right.

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Mother Functionrelative functionchange? y 1 = cos x y 2 = 2 + cos x vertical translation, 2 units up. y 1 = cos xy 2 = -3 + cos x vertical translation, 3 units down. 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 Functionrelative functionchange? y 1 = sin x y 2 = sin 2x y 2 = sin x Period = 180 or Period = 720 or generalization? y = sin bx Period = or

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y = d + a sin (bx - c) y = d + a cos (bx - c) a is the amplitude is the horizontal translation d is the vertical translation period = Summary: or

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Analyze the graph of amplitude = vertical translation: horizontal translation: period = none

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Analyze the graph of amplitude = vertical translation: horizontal translation: period = none 3 (to the left)

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Analyze the graph of amplitude = vertical translation: horizontal translation: period = Up 2 3 none

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

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The most important thing to remember about graphing is determining the starting point and the stopping point on the t-table. You must know how to analyze the equation before you can graph it.

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y = sin (2 + ) amplitude = period = vertical translation: horizontal translation: xy ) horiz. tells you where to start 2) add the period to find out where to finish 3) divide period by 4 to find increments 4) plot points and graph Ex #6cGraph = 3 up 1 table goes in increments of mid high low

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