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

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

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. -1 1 sin x x 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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Sine Function

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

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

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

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

Example: Sketch the graph of y = 3 cos x on the interval [–, 4]. Partition the interval [-π,4] on your x-axis max x-int min 3 -3 y = 3 cos x 2  x y x (0, 3) ( , 3) ( , 0) ( , 0) ( , –3) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: y = 3 cos x

If |a| > 1, the amplitude stretches the graph vertically. 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 = 2sin x y = sin x y = sin x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Amplitude

If k > 1, the graph of the function is shrunk horizontally. 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 . For k  0, the period of y = a tan kx is . If k > 1, the graph of the function is shrunk horizontally. y x period: period: 2 If 0 < k < 1, the graph of the function is stretched horizontally. y x period: 4 period: 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Period of a Function

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

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

The Graph of y = Asin(Kx - C) 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. y y = A sin Kx Amplitude: | A| x Starting point: x = -C/K Period: 2/B Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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

Phase Shift: -c/k (opposite sign) Shift horizontally left and right Amplitude Period: 2π/k Phase Shift: -c/k (opposite sign) Shift horizontally left and right Vertical Shift up and down Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

State the periods of each function: 1. 2. 4π or 720° π/2 or 90°

State the phase shift of each function: 1. 2. Right phase shift 45° Left phase shift -90°

State the amplitude, period, and phase and Vertical shift of each function: 1. 2. 3. 4. 5. 6. A = 4, period = 360°, Phase shift = 0° Vertical Shift = none A = 4, period = 720°, Phase shift = 0° Vertical Shift = none A = NONE, period = 45°, Phase shift = 0° Vertical Shift = none A = NONE, period = 90°, Phase shift = π/2 Right Vertical Shift = none A = 2, period = 180°, Phase shift = 0° Vertical Shift = none A = 3, period = 360°, Phase shift = 90° Right Vertical Shift = none

State the amplitude, period, and phase and vertical shift of each function: 1. 2. A = 10, period = 1080°, Phase shift = 900° Right Vertical Shift = none A = 243, period = 24°, Phase shift = 8/3° Vertical Shift = none

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°

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