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Graphing Sine and Cosine Functions

Graphing 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 . Graphing Sine and Cosine Functions. Objectives. Key words.

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Graphing Sine and Cosine Functions

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  1. Graphing 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. Graphing Sine and Cosine Functions Objectives Key words Graph the equations of sine and cosine functions given the amplitude, period, phase shift, and vertical translation Write equations given a graph. Graph compound functions Midline Amplitude Maximum Minimum Period Sine curve Cosine curve Phase shift

  3. Quick check! Can you find the distance between two numbers? Can you find the midpoint between two numbers?

  4. 1: Graphing Sine and Cosine Functions Order does matter! y=A sin⁡[B(θ-h)]+k y=A cos⁡[B(θ-h)]+k Draw the vertical shift, k, and graph the midline y=k. Use a solid line. Draw the amplitude, . Use dashed lines to indicate the maximum and minimum values of the function. Draw the period of the function, , and graph the appropriate sine or cosine curve. Draw the phase shift, h, and translate the graph accordingly.

  5. 1: Graphing Sine and Cosine Functions State the amplitude, period, phase shift, and vertical shift for y= 4cos(x / 2 + π) - 6. Then graph the function.

  6. 1: Graphing Sine and Cosine Functions State the amplitude, period, phase shift, and vertical shift for y= 4cos(x / 2 + π) - 6. Then graph the function. Amplitude is 4 Period is 4π Phase shift is -2π Vertical shift is -6

  7. 1: Graphing Sine and Cosine Functions State the amplitude, period, phase shift, and vertical shift for y= 2cos(x / 4 + π) - 1. Then graph the function.

  8. 1: Graphing Sine and Cosine Functions State the amplitude, period, phase shift, and vertical shift for y= 2cos(x / 4 + π) - 1. Then graph the function. Amplitude is 2 Period is 8π Phase shift is -4π Vertical shift is -1

  9. 2: Write Equations of Sine and Cosine Order does matter! y=A sin⁡[B(θ-h)]+k y=A cos⁡[B(θ-h)]+k Determine the vertical shift, k, from the midline y=k. Determine the amplitude, . From the maximum and minimum values of the function. Determine the period of the function, , from one complete interval. Determine the phase shift, h, from either sine and/or cosine.

  10. 2: Write Equations Example State the amplitude, period, phase shift, and vertical shift for the graph of:

  11. 2: Write Equations Example State the amplitude, period, phase shift, and vertical shift for the graph of: The amplitude is 2 or 2. The period is or 4. The phase shift is or -2. The vertical shift is +3 y = 2 cos( /2 + ) + 3 or y = 2 cos(1/2( + 2))+ 3

  12. 2: Write Equations Example YOU TRY! State the amplitude, period, phase shift, and vertical shift for the graph of:

  13. 2: Write Equations Example YOU TRY! State the amplitude, period, phase shift, and vertical shift for the graph of: Vertical shift is 0, midline y=0 Amplitude is 3 Period is 2π/3 Phase shift is π/3 f(x) = 3cos(3x + π)

  14. 3: Graph Compound Functions Types of Compound Functions For Example: Compound functions may consist of sums or products of trigonometric functions or other functions. • Product of trigonometric functions • Sum of a trigonometric function and a linear function.

  15. 3: Graph Compound Functions Graph y = x + sin x.

  16. 3: Graph Compound Functions Graph y = x + sin x. First create a table of each graph: y= x or y= sin x

  17. 3: Graph Compound Functions YOU TRY: Graph y = x + cosx. First create a table of each graph: y = x or y = cosx

  18. 3: Graph Compound Functions YOU TRY: Graph y = x + cosx. First create a table of each graph: y = x or y = cosx

  19. Conclusion Summary Assignment Now you know how to graph sinusoidal functions Ask questions while you finish the assignment Finish missing work Exam Thursday/Friday • 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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