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Section 4.5 Graphs of Sine and Cosine

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Sine Curve Key Points:0 Value: 0 100 22 22 1

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Cosine Curve Key Points:0 Value: 1 0 0 1 22 22 1

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Equations For the rest of this section, we will be graphing: y = a Sin (bx c) + d y = a Cos (bx c) + d y = Sin x a = 1 b = 1 c = 0 d = 0

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Graph the equation y = 2 Sin x 22 1 -2 Key Points:0 Value: 0 20-20 22 2

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Amplitude (a) Half the distance between the maximum and minimum values of the function Given by the value of a Graph the functions: y = 4 Sin x y = Cos x y = -2 Sin x

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22 4 -4 y = 4 Sin x y = Cos x 3 2 1 -2 -3 y = -2Sin x

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y = a Sin (bx c) + d b gives us the period of the curve Period = y = 4 Sin 2x Amplitude = Period = 4 =

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Key Points Would having a period of change the key points of the curve? 22 1

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Finding Key Points In GeneralFor Y = 4Sin 2x 1) Find the period of the curve 2) Divide the period into 4 equal parts 3) From your starting point, add this distance 4 times for each period 1) Period = 2) Distance = 3) 0,,,,

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1 y = 4Sin 2x 4 -4

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Graph the following curves y = 4 Cos 8x y = Cos 2 x y = -2 Sin 6x

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y = 4Cos 8x Amplitude =4b =8 Period = Distance = 4 -4

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y = Cos 2 x Amplitude =b = 22 Period = Distance = -

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y = -2Sin 6x Amplitude =2b =6 Period = Distance = 2 - 2

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y = a Sin (bx c) + d a = b = c = amplitude Find the period Find the phase shift horizontal shift

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y = Sin (x - ) a = b = c = 1 Period = P. S. =

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y = -3 Cos (2 x + 4 ) a = b = c = 3 22 Period = P. S. =

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y = a Sin (bx c) + d a = b = c = d = amplitude Find the period Find the phase shift Vertical Shift

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y = a = b = c = d = 2 Period = P. S. = 3

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y = a = b = c = d = 4 Period = P. S. = -2

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1 4 2 -6 -2 y = -2