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Amplitude and Period of Sine and Cosine Functions

Amplitude and Period 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. Recap of Last week.

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Amplitude and Period of Sine and Cosine Functions

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  1. Amplitude and Period 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. Recap of Last week • Change radians to degrees and vice versa • Arc Length • Sector Area • Determine whether the function is periodic. If so, state the period • Find sin⁡(-5π) by referring to the graph of the sine function. • Find the values of θ for which cos⁡(θ)=1 is true. • Graph the function y=sin⁡(x) for the interval -12π≤x≤-10π. • Determine whether the graph is y=sin⁡(x), y=cos⁡(x), or neither.

  3. Take 30 Minutes • Finish the assignments you were given last week • The assignments will posted shortly on the projector from the website • If you are finished, see the teacher for further information

  4. Objectives • Find the amplitude and period for sine and cosine functions. • Write equations of sine and cosine functions given the amplitude and period.

  5. General Information you NEED to know The summary of Transformations for the sinusoidal functions: • and • The amplitude is • The period is • The horizontal shift is • The midline is • is the angular frequency; that is, the number of cycles completed in

  6. Objective 1: The Amplitude The amplitude is • If , then we have a vertical compression • If , then we have a vertical expansion • A negative A is a reflection about the x-axis

  7. Example 1: State Amplitude • a. • State the amplitude for the function y = 3 cos. • b. • Graph y = 3 cos and y = cos on the same set of axes. • c. • Compare the graphs.

  8. Example 1: State Amplitude • A. According to the definition of amplitude, the amplitude of y = Acos is A. So, the amplitude of y= 3 cos is 3 or 3.

  9. Example 1: State Amplitude • B.Make a table of values. Then graph the points and draw a smooth curve.

  10. C. Both graphs cross the axis at the same points and also reach the minimum and maximum values at the same points. The difference is that the minimum and maximum values of y= cos are -1 and 1, and the minimum and maximum values of y = 3 cos are –3 and 3.

  11. Objective 1: The Period The period is • If , then we have a horizontal expansion • If , then we have a horizontal compression • A negative B is a reflection about the y-axis

  12. Example 1: State the Period • a. State the period for the function y = cos. • b. Graph y = cos and y = cos.

  13. Example 2: State the Period • a. The definition of the period of y = cosk is . So, the period for the function y= cos is or 8. • b.

  14. Extra Example: • State the amplitude and period for the function y= sin 2. Then graph the function.

  15. Extra Example: • Since A = , the amplitude is the absolute value of Aor . Since k = 2, the period is or . • Use the basic shape of the sinefunction and the amplitude and period to graph the equation.

  16. Objective 2: Write Equation Examples Write an equation of the cosine function with amplitude 4.5 and period 8. • The form of the equation will be y = Acosk. First, find the possible values of Afor an amplitude of 4.5. A = 4.5 A = 4.5 or –4.5 • Since there are two values of A, two possible equations exist. • Now find the value of k when the period is 8. = 8The period of a cosine function is . k = or • The possible equations are y = 4.5 cos or y = -4.5 cos.

  17. Real-World Application PHYSICS The motion of a weight on a certain kind of spring can be described by a modified trigonometric function. At time 0, Carrie pushes the weight upward 3 inches from its equilibrium point and releases it. She finds that the weight returns to the point three inches above the equilibrium point after 2 seconds. a. Write an equation for the motion of the weight. b. What will the position of the weight be after 15.5 seconds? • a.At time 0, the weight is 3 inches above the equilibrium point and at its maximum value. Since the weight will fall, the values will get smaller. The function will be a cosine function with a positive value of A. • The maximum and minimum points are 3 and –3. Thus, the amplitude, A, is 3. • The weight makes a complete cycle is 2 seconds. Thus, the period is 2. = 2 k =  • Now, write the equation. y = Acoskt y = 3 cost • b.Use a calculator to find the value of y when t = 15.5 to find the position of the weight after 15.5 seconds. y = 3 cos (15.5) y = 0 • After 15.5 seconds, the weight will be at the equilibrium point.

  18. Conclusion Summary Assignment • Grab a partner: • Have one student explain to other students how to alter the period and amplitude for the basic sine graph and/or cosine graph. • 6.4 Amplitude and Period of Sine and Cosine Functions • pg373#(17-22 ALL, 23-51 ODD, 57,60 EC). • Problems not finished will be left as homework.

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