#### Transcript PC 01-29n30 Graphing Sine and Cosine Functions

## 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.