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Translations of Sine and Cosine Functions

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.

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Translations of Sine and Cosine Functions

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  1. 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.

  2. 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 0.27 and period π/2. • Write an equation of the sine function with amplitude 3/5 and period 4. Answers: 3, π2/3, 8π1, π/2 y=±0.27 sin⁡(4θ) y=±3/5 cos⁡(π/2 θ)

  3. 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.

  4. 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

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

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

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

  8. 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

  9. 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.

  10. 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.

  11. Additional Information for Graphing 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.

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

  13. 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 ory = - 5 sin (2/3 - /3) + 2

  14. Objective 4: Graph Compound Functions • Compound functions may consist of sums or products of trigonometric functions or other functions. • For example: • Product of trigonometric functions • Sum of a trigonometric function and a linear function.

  15. 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.

  16. Conclusion Summary Assignment • A Japanese company invented the first integrated radio circuit in 1966. 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, 21-37 ODD, 42,45 EC) • Problems not finished will be left as homework.

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