Presentation on theme: "Translations of Sine and Cosine Functions"— Presentation transcript:
1Translations 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.
2Quick CheckState 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)
3ObjectivesFind 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 translationGraph compound functions.
4Objective 1: Phase shift Phase shift of Sine and Cosine Functions:y=A sin[B(θ-h)]+k and y=A cos[B(θ-h)]+kThe horizontal shift is hIf h>0, the shift is to the rightIf h<0, the shift is to the left
5Objective 1: Phase Shift Example State the phase shift for each function. Then graph the function.a. y = sin (2 + )b. y = cos ( - )
6A. 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 .
7B. 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 .
8Objective 2: Vertical shift Vertical shift of Sine and Cosine Functions:y=A sin[B(θ-h)]+k and y=A cos[B(θ-h)]+kThe midline is y=kIf k>0, the shift is upwardIf k<0, the shift is downward
9Objective 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.
10The 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 functionis 3, draw dashed lines parallel to the midline which are 3 units above and below the midline. Then draw the cosine curve.
11Additional 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.
12Additional 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 2𝜋 or 4.The phase shift is − 𝜋 or -2. The vertical shift is +3
13Objective 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| = 5A = 5 or -5k: 2/k = 3 The period is 3.k = 2/3c: -c/k = /2 The phase shift is /2.-c/ 2/3 = /2 k = 2/3c = - /3h: 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
14Objective 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 functions𝑦= cos 𝑥 +𝑥 Sum of a trigonometric function and a linear function.
15Objective 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.xsin xx + sin x/21+ 1 2.57 3.143/2-1- 1 3.7122 6.285/2+ 1 8.8533 9.42
16Conclusion 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 Functionspg383#(14-20 ALL, ODD, 42,45 EC)Problems not finished will be left as homework.