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Amplitude, Period, & Phase Shift 6.2 Trig Functions 3 ways we can change our graphs

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Properties of Sine and Cosine Functions 6. The cycle repeats itself indefinitely in both directions of the x-axis. Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 1. The domain is the set of real numbers. 5. Each function cycles through all the values of the range over an x-interval of. 2. The range is the set of y values such that.

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Sine Function Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 0010sin x 0x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = sin x

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θtan θ −π/2 −∞−∞ −π/4−1 00 π/41 π/2 ∞ 0 θ tan θ −π/2 π/2 One period: π 3π/2 −3π/2 Vertical asymptotes where cos θ = 0 Graph of Tangent Function: Periodic

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θtan θ 0 ∞ π/41 π/20 3π/4−1 π −∞ 3π/2 −3π/2 Vertical asymptotes where sin θ = 0 Graph of Cotangent Function: Periodic π -π-π −π/2 π/2 cot θ

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Cosecant is the reciprocal of sine One period: 2π π 2π2π 3π3π 0 −π −2π −3π Vertical asymptotes where sin θ = 0 θ csc θ sin θ

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Secant is the reciprocal of cosine One period: 2π π 3π3π −2π 2π2π −π −3π 0 θ sec θ cos θ Vertical asymptotes where cos θ = 0

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 y x Example: y = 3 cos x Example: Sketch the graph of y = 3 cos x on the interval [– , 4 ]. Partition the interval [-π,4 ] on your x-axis maxx-intminx-intmax y = 3 cos x 22 0x (0, 3) (, 0) (, 3) (, –3)

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Amplitude The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically. If 0 < |a| < 1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis. y x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x y = 2sin x y = sin x

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 y x Period of a Function period: 2 period: The period of a function is the x interval needed for the function to complete one cycle. For k 0, the period of y = a sin kx is. For k 0, the period of y = a cos kx is also. If 0 < k < 1, the graph of the function is stretched horizontally. If k > 1, the graph of the function is shrunk horizontally. y x period: 2 period: 4 For k 0, the period of y = a tan kx is.

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 y x y = cos (–x) Graph y = f(-x) Use basic trigonometric identities to graph y = f (–x) Example 1: Sketch the graph of y = sin (–x). Use the identity sin (–x) = – sin x The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. Example 2: Sketch the graph of y = cos (–x). Use the identity cos (–x) = – cos x The graph of y = cos (–x) is identical to the graph of y = cos x. y x y = sin x y = sin (–x) y = cos (–x)

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 y x 0 20 –2 0y = –2 sin 3x 0 x Example: y = 2 sin(-3x) Example: Sketch the graph of y = 2 sin (–3x). Rewrite the function in the form y = a sin kx with k > 0 amplitude: |a| = |–2| = 2 Calculate the five key points. (0, 0) (, 0) (, 2) (, -2) (, 0) Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x period: =

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 The graph of y = A sin (Kx – C) is obtained by horizontally shifting the graph of y = A sin Kx so that the starting point of the cycle is shifted from x = 0 to x = - C / K. The number – C / K is called the phase shift. amplitude = | A| period = 2 / K. The graph of y = A sin (Kx – C) is obtained by horizontally shifting the graph of y = A sin Kx so that the starting point of the cycle is shifted from x = 0 to x = - C / K. The number – C / K is called the phase shift. amplitude = | A| period = 2 / K. x y Amplitude: | A| Period: 2 /B y = A sin Kx Starting point: x = -C/K The Graph of y = Asin(Kx - C)

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Example Determine the amplitude, period, and phase shift of y = 2sin(3x- ) Solution: Amplitude = |A| = 2 period = 2 /K = 2 /3 phase shift = -C/K = /3 to the right

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Example cont. y = 2sin(3x- )

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Amplitude Period: 2π/k Phase Shift: -c/k Vertical Shift

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17 State the periods of each function: π or 720° π/2 or 90°

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18 State the phase shift of each function: Right phase shift 45° Left phase shift -90°

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19 State the amplitude, period, and phase shift of each function: A = 4, period = 360°, Phase shift = 0° A = NONE, period = 45°, Phase shift = 0° A = 2, period = 180°, Phase shift = 0° A = 4, period = 720°, Phase shift = 0° A = NONE, period = 90°, Phase shift = π/2 Right A = 3, period = 360°, Phase shift = 90° Right

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20 State the amplitude, period, and phase shift of each function: A = 10, period = 1080°, Phase shift = 900° Right A = 243, period = 24°, Phase shift = 8/3°

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21 Write an equation for each function described: 1.) a sine function with amplitude 7, period 225°, and phase shift -90° 2.) a cosine function with amplitude 4, period 4π, and phase shift π/2 3.) a tangent function with period 180° and phase shift 25°

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 Graph each function: 1.) 2.)

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