## Graphing Sine and Cosine Functions

T R I G O N O M E T R Y , 4 . 0 : S T U D E N T S G R A P H F U N C T I O N S O F T H E F O R M F ( T ) = A S I N ( B T + C ) O R F ( T ) = A C O S ( B T + C ) A N D I N T E R P R E T A , B , A N D C I N T E R M S O F A M P L I T U D E , F R E Q U E N C Y , P E R I O D , A N D P H A S E S H I F T .

Graphing Sine and Cosine Functions

Objectives

1.

2.

3.

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

Key words

        Midline Amplitude Maximum Minimum Period Sine curve Cosine curve Phase shift

Quick check!

 Can you find the distance between two numbers?

 Can you find the midpoint between two numbers?

1: Graphing Sine and Cosine Functions

Order does matter!

1.

2.

Draw the vertical shift, k, and graph the midline y=k. Use a solid line.

Draw the amplitude, 𝐴 . Use dashed lines to indicate the maximum and minimum values of the function.

y=A sin[B(θ-h)]+k y=A cos[B(θ-h)]+k

3.

4.

Draw the period of the function, 2𝜋 𝐵 , and graph the appropriate sine or cosine curve.

Draw the phase shift, h, and translate the graph accordingly.

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.

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

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.

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

2: Write Equations of Sine and Cosine

Order does matter!

1.

2.

Determine the vertical shift, k, from the midline y=k.

Determine the amplitude, the function.

𝐴 . From the maximum and minimum values of

y=A sin[B(θ-h)]+k y=A cos[B(θ-h)]+k

3.

4.

Determine the period of the function, interval.

2𝜋 𝐵 , from one complete Determine the phase shift, h, from either sine and/or cosine.

2: Write Equations Example

State the amplitude, period, phase shift, and vertical shift for the graph of:

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 2𝜋 1 or 4  . 2 The phase shift is − 𝜋 1 or -2  . The vertical shift is +3 y = 2 cos (  2 /2 +  ) + 3 or y = 2 cos (1/2(  + 2  )) + 3

2: Write Equations Example

YOU TRY! State the amplitude, period, phase shift, and vertical shift for the graph of:

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 2 π/3 Phase shift is π/3 f(x) = 3cos(3x + π)

3: Graph Compound Functions

Types of 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.

3: Graph Compound Functions

Graph y = x + sin x.

3: Graph Compound Functions

Graph y = x + sin x.

 3 5  First create a table of each graph: y = x or y = sin x  2   x 0 /2  /2 /2

sin x x + sin x

0 0 1 0 -1 0

1

 /2 + 1 2.57

 3.14

3  /2 - 1 3.71

2  6.28

5  /2

+ 1 8.85

3: Graph Compound Functions

YOU TRY: Graph y = x + cos x.

 First create a table of each graph: y = x or y = cos x

3: Graph Compound Functions

YOU TRY: Graph y = x + cos x.

 5 First create a table of each graph: y = x or y = cos x  x 0  /2  3  /2 2  /2

cos x x + cos x

1 1 0 -1 0 1

0

 /2  1.57

-1 2.14

3  /2 2  +1 7.28

5  /2 4.71

7.85

Conclusion

Summary

    Now you know how to graph sinusoidal functions Ask questions while you finish the assignment Finish missing work Exam Thursday/Friday

Assignment

 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.