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4.5 Sinusoidal Graphs Sketching and Writing Equations

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Sinusoidal Graphs Periodic Function Graphs of sine and cosine functions. Functions whose graphs have a repeating pattern. Period The horizontal length of each cycle in a periodic graph. Midline The horizontal line halfway between a sinusoid’s max/min values.

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The Graph of y = sin x x y

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Characteristics of the Sine Function The domain is the set of all real numbers. The range consists of all real numbers from -1 to 1, inclusive. The sine function is an odd function (symmetric with respect to the origin). The sine function has a period of 2π Midline is the x-axis

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Period 3 2 1 -2 -3 y = sinx 1 period pattern starts Period= 2 pattern starts 1 period

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The Graph of y = cos x x y

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Characteristics of the Cosine Function The domain is the set of all real numbers. The range consists of all real numbers from -1 to 1, inclusive. The cosine function is an even function (symmetric with respect to the y-axis). The cosine function has a period of 2π Midline is the x-axis

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Period 3 2 1 -2 -3 1 period y = cosx pattern starts Period = 2 pattern starts 1 period

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Period 3 2 1 -2 -3 1 period pattern starts Period=

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Period 3 2 1 -2 -3 1 period pattern starts Period = 33 2

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Amplitude Half the distance between the maximum and minimum of a sinusoidal graph.

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Amplitude 3 2 1 -2 -3 Amplitude = 1 Amplitude max. min.

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Amplitude 3 2 1 -2 -3 max. min. Amplitude Amplitude = 2

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Amplitude 3 2 1 -2 -3 max. min. Amplitude Amplitude = 1

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Amplitude 3 2 1 -2 -3 max. min. Amplitude Amplitude = 2 33

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Sinusoidal Graphs Amplitude = Period = General Form of Equations

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Sinusoidal Graphs y = 3sin4x Amplitude = Period = y = -4cos5x Amplitude = Period =

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Writing Equations Answer these questions first: 1.Is it a “sin” or “cos” graph? 2.What is the amplitude? 3.What is the period? 4.Is the graph reflected? 5.Is it shifted up/down? 6.Is it shifted left/right?

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Writing Equations b = Write an equation: cosine graph amplitude = 4 Period = a =4 b == 2

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Writing Equations b = Write an equation: sine graph amplitude = 3 Period = 4 a =3 b =

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Writing Equations b = Write an equation: sine graph amplitude = 2 Period = 3 Reflected over x-axis a =±2±2 b =

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Writing Equations b = a = b = Write an equation: cosine graph amplitude = ½ Period = Reflected over x-axis Invert and multiply!

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Writing Equations Write an Equation Sine graph Amplitude = 5 Period = Shifted up 2 a = b = Changing the midline causes vertical shifts

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Sinusoidal Graphs 3 2 1 -2 -3 Period = Amplitude = b = y = sin2x “sin” 1 a =1

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Sinusoidal Graphs 3 2 1 -2 -3 Period = Amplitude b = y = 3sinx “sin” 22 = 3 a =3

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Sinusoidal Graphs 3 2 1 -2 -3 Period = Amplitude = “cos” b = 1 a =1

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Sinusoidal Graphs 3 2 1 -2 -3 Period = Amplitude = “cos” b = 2 33 Reflected over x-axis! a = -2

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Sinusoidal Graphs 3 2 1 -2 -3 Period = Amplitude = “cos” b = 3 6 Reflected over x-axis! a =-3

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Graphing Sinusoidal Functions y = 3sin4x 3 2 1 -2 -3

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Graphing Sinusoidal Functions y = 2sin x 3 2 1 -2 -3

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Graphing Sinusoidal Functions y = -3sin2x 3 2 1 -2 -3

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Graphing Sinusoidal Functions 3 2 1 -2 -3

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Graphing Sinusoidal Functions 3 2 1 -2 -3

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Graphing Sinusoidal Functions 3 2 1 -2 -3

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Phase Shift Horizontal translations of sine and cosine graphs. Left Right Phase Shift

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Amp =3 T = P.S. = right

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Phase Shift Amp =2 T = P.S. = left

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Phase Shift “Cosine” Amplitude = 5 Period = Phase shift =

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Phase Shift “sine” Amplitude = 3 Period = 4 Phase shift =

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