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.

Section 4.5: Figure 4.49, Key Points in the Sine and Cosine Curves

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

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

The Graph of y = AsinB(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. amplitude = | A| period = 2 /B. y y = A sin Bx Amplitude: | A| x Starting point: x = C Period: 2/B Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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 = 2 sin x y = sin x y = sin x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x Amplitude

If b > 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 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 x period: period: 2 If b > 1, the graph of the function is shrunk horizontally. y x period: 4 period: 2 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) Graph y = f(-x)

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

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 = 4sin  y = 2cos1/2  y = sin (4x - ) Amplitude: Phase shift: Period:

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. max x-int min 3 -3 y = 3 cos x 2  x y x (0, 3) ( , 3) ( , 0) ( , 0) ( , –3) Example: y = 3 cos x

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 = 3sin  period is 90. use P = rewrite the eq. y = 3 sin4 45 degrees left means +45 Answer: y = 3sin4( + 45)

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 = 1/2cos x period is /4. use P = rewrite the eq. y = 1/2cos 8x right  is negative, put this phase shift inside the parenthesis w/ opposite sign. Answer: y = 1/2cos8(x - )