Amplitude Period and Phase Shift Objectives I can

Amplitude, Period, and Phase Shift

Objectives • I can determine amplitude, period, and phase shifts of trig functions • I can write trig equations given specific period, phase shift, and amplitude. 2

Section 4. 5: Figure 4. 49, Key Points in the Sine and Cosine Curves 3

Radian versus Degree • We will use the following to graph or write equations: – “x” represents radians – “ ” represents degrees – Example: sin x versus sin 4

Amplitude Period: 2π/b Phase Shift: Left (+) Right (-) Vertical Shift Up (+) Down (-) 5

The Graph of y = Asin. B(x - C) The graph of y = A sin B(x – C) is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the cycle is shifted from x = 0 to x = C. The number C is called the phase shift. y amplitude = | A| period = 2 /B. y = A sin Bx Amplitude: | A| x Starting point: x = C Period: 2 /B Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6

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 7

The period of a function is the x interval needed for the function to complete one cycle. For b 0, the period of y = a sin bx is . For b 0, the period of y = a cos bx is also . If 0 < b < 1, the graph of the function is stretched horizontally. y period: 2 period: x If b > 1, the graph of the function is shrunk horizontally. y period: 2 x period: 4 8

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) 9

Example Determine the amplitude, period, and phase shift of y = 2 sin (3 x - ) Solution: First factor out the 3 y = 2 sin 3(x - /3) Amplitude = |A| = 2 period = 2 /B = 2 /3 phase shift = C/B = /3 right 10

Find Amplitude, Period, Phase Shift • Amplitude (the # in front of the trig. Function • Period (360 or 2 divided by B, the #after the trig function but before the angle) • Phase shift (the horizontal shift after the angle and inside the parenthesis) • y = 4 sin y = 2 cos 1/2 y = sin (4 x - ) Amplitude: Phase shift: Period: 11

Example: Sketch the graph of y = 3 cos x on the interval [– , 4 ]. Partition the interval [0, 2 ] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. 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) 12

Writing Equations • Write an equation for a positive sine curve with an amplitude of 3, period of 90 and Phase shift of 45 left. • Amplitude goes in front of the trig. function, write the eq. so far: • y = 3 sin • period is 90. use P = • • rewrite the eq. • y = 3 sin 4 • 45 degrees left means +45 • Answer: y = 3 sin 4( + 45) 13

Writing Equations • Write an equation for a positive cosine curve with an amplitude of 1/2, period of and Phase shift of right . • Amplitude goes in front of the trig. function, write the eq. so far: • y = 1/2 cos x • period is /4. use P = • • rewrite the eq. • y = 1/2 cos 8 x • right is negative, put this phase shift inside the parenthesis w/ opposite sign. • Answer: y = 1/2 cos 8(x - ) 14

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