6 2 Trig Functions Amplitude Period Phase Shift

6. 2 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. 2

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. x 0 sin x 0 1 0 -1 0 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 = sin x y x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3

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

Example: Sketch the graph of y = 3 cos x on the interval [– , 4 ]. Partition the interval [-π, 4 ] on your x-axis x y = 3 cos x (0, 3) y 0 3 max 0 -3 0 2 3 x-int min x-int max ( , 3) x ( ( , 0) ( , – 3) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5

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 y = 2 sin x x y= sin x y = – 4 sin x reflection of y = 4 sin x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. y = sin x y = 4 sin x 6

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 they function is shrunk horizontally. period: 2 period: x If 0 < k < 1, the graph of the function is stretched horizontally. y period: 2 x period: 4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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 = sin (–x) y Use the identity sin (–x) = – sin x 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 Use the identity x cos (–x) = – cos x y = cos (–x) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8

Example: Sketch the graph of y = 2 sin (– 3 x). Rewrite the function in the form y = a sin kx with k > 0 y = 2 sin (– 3 x) = – 2 sin 3 x Use the identity sin (– x) = – sin x: period: 2 = 2 amplitude: |a| = |– 2| = 2 3 Calculate the five key points. x 0 y = – 2 sin 3 x 0 y – 2 0 ( , 2) x (0, 0) ( , -2) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9

Amplitude Period: 2π/k Phase Shift: -c/k Vertical Shift Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10

Example Determine the amplitude, period, and phase shift of y = 2 sin(3 x- ) 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. 11

Example cont. • y = 2 sin(3 x- ) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12

State the periods of each function: 1. 4π or 720° 2. π/2 or 90° 13

State the phase shift of each function: 1. Right phase shift 45° 2. Left phase shift -90° 14

State the amplitude, period, and phase shift of each function: 1. 4. A = 4, period = 360°, Phase shift = 0° 2. A = 4, period = 720°, Phase shift = 0° 5. A = NONE, period = 45°, Phase shift = 0° 3. A = 2, period = 180°, Phase shift = 0° A = NONE, period = 90°, Phase shift = π/2 Right 6. A = 3, period = 360°, Phase shift = 90° Right 15

State the amplitude, period, and phase shift of each function: 1. A = 10, period = 1080°, Phase shift = 900° Right 2. A = 243, period = 24°, Phase shift = 8/3° 16

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° 17

Graph each function: 1. ) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2. ) 18