# Translations of Sine and Cosine Functions

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Translations of Sine and Cosine Functions
Trigonometry, 4.0: Students graph functions of the form f(t)=Asin(Bt+C) or f(t)=Acos(Bt+C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift.

Quick Check State the amplitude and period for each function. Then graph each function. y=-3 cos⁡(2θ) y=2/3 cos⁡(θ/4) y=sin⁡(4θ) Write an equation of the sine function with amplitude and period π/2. Write an equation of the sine function with amplitude 3/5 and period 4. Answers: 3, π 2/3, 8π 1, π/2 y=±0.27 sin⁡(4θ) y=±3/5 cos⁡(π/2 θ)

Objectives Find the phase shift translations for sine and cosine functions. Find the vertical translations for sine and cosine functions. Write the equations of sine an cosine functions given the amplitude, period, phase shift, and vertical translation Graph compound functions.

Objective 1: Phase shift
Phase shift of Sine and Cosine Functions: y=A sin⁡[B(θ-h)]+k and y=A cos⁡[B(θ-h)]+k The horizontal shift is h If h>0, the shift is to the right If h<0, the shift is to the left

Objective 1: Phase Shift Example
State the phase shift for each function. Then graph the function. a. y = sin (2 + ) b. y = cos ( - )

A. y = sin (2 + ) The phase shift of the function is − 𝑐 𝑘 or − 𝜋 2
A. y = sin (2 + ) The phase shift of the function is − 𝑐 𝑘 or − 𝜋 To graph y = sin (2 + ), consider the graph of y = sin 2. Graph this function and then shift the graph − 𝜋 2 .

B. y = cos ( - ) The phase shift of the function is − 𝑐 𝑘 or − 𝜋 1 , which equals . To graph y = cos ( - ), consider the graph of y = cos  and then shift the graph .

Objective 2: Vertical shift
Vertical shift of Sine and Cosine Functions: y=A sin⁡[B(θ-h)]+k and y=A cos⁡[B(θ-h)]+k The midline is y=k If k>0, the shift is upward If k<0, the shift is downward

Objective 2: Vertical shift Example
State the vertical shift and the equation of the midline for the function y = 3 cos  + 4. Then graph the function.

The vertical shift is 4 units upward. The midline is the graph y = 4.
To graph the function, draw the midline, the graph of y = 4. Since the amplitude of the function is 3, draw dashed lines parallel to the midline which are 3 units above and below the midline. Then draw the cosine curve.

Graphing Sine and Cosine Functions: Determine the vertical shift and graph the midline. Determine the amplitude. Use dashed lines to indicate the maximum and minimum values of the function. Determine the period of the function and graph the appropriate sine or cosine curve. Determine the phase shift and translate the graph accordingly.

State the amplitude, period, phase shift, and vertical shift for y = 2 cos ( /2 + ) + 3. The amplitude is 2 or 2. The period is 2𝜋 or 4. The phase shift is − 𝜋 or -2. The vertical shift is +3

Objective 3: Write Equation Example
Write an equation of a sine function with amplitude 5, period 3, phase shift /2, and vertical shift 2. The form of the equation will be y = A sin (k + c) + h. Find the values of A, k, c, and h. A: |A| = 5 A = 5 or -5 k: 2/k = 3 The period is 3. k = 2/3 c: -c/k = /2 The phase shift is /2. -c/ 2/3 = /2 k = 2/3 c = - /3 h: h = 2  Substitute these values into the general equation. The possible equations: y = 5 sin (2/3 - /3) + 2 or y = - 5 sin (2/3 - /3) + 2

Objective 4: Graph Compound Functions
Compound functions may consist of sums or products of trigonometric functions or other functions. For example: 𝑦= sin 𝑥 ∙ cos 𝑥 Product of trigonometric functions 𝑦= cos 𝑥 +𝑥 Sum of a trigonometric function and a linear function.

Objective 4: Graphing Example
Graph y = x + sin x. First graph y = x and y = sin x on the same axes. Then add the corresponding ordinates of the functions. Finally, sketch the graph. x sin x x + sin x /2 1 + 1  2.57   3.14 3/2 -1 - 1  3.71 2 2  6.28 5/2 + 1  8.85 3 3  9.42

Conclusion Summary Assignment
A Japanese company invented the first integrated radio circuit in Suppose that researchers were observing a sine curve that had an amplitude of 3 centimeters, a period of 9 centimeters, an upper shift of 2 centimeters, and a phase shift ½ centimeter to the right. State the function that models the data. y=3 sin⁡(2π/9 θ - π/9) + 2, this is one of many equivalent answers possible. 6.5 Translations of Sine and Cosine Functions pg383#(14-20 ALL, ODD, 42,45 EC) Problems not finished will be left as homework.