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

1. Graph the equations of sine and cosine functions given the amplitude, period, phase shift, and vertical translation

2. Write equations given a graph.

3. Graph compound functions

MidlineAmplitudeMaximumMinimumPeriodSine curveCosine curvePhase shift

Graphing Sine and Cosine Functions

Quick check!

Can you find the distance between two numbers?

Can you find the midpoint between two numbers?

Order does matter!y=A sin [B(θ-h)]+ky=A cos [B(θ-h)]+k

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

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

3. Draw the period of the function, , and graph the appropriate sine or cosine curve.

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

1: Graphing Sine and Cosine Functions

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 4Period is 4πPhase shift is -2πVertical 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 2Period is 8πPhase shift is -4πVertical shift is -1

Order does matter!y=A sin [B(θ-h)]+ky=A cos [B(θ-h)]+k

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

2. Determine the amplitude, . From the maximum and minimum values of the function.

3. Determine the period of the function, , from one complete interval.

4. Determine the phase shift, h, from either sine and/or cosine.

2: Write Equations of Sine and Cosine

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 or 4. The phase shift is or -2. The vertical shift is

+3y = 2 cos ( /2 + ) + 3 ory = 2 cos (1/2( + 2)) + 3

2: Write Equations Example

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=0Amplitude is 3Period is 2 π/3Phase shift is π/3f(x) = 3cos(3x + π)

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.

3: Graph Compound Functions

Graph y = x + sin x.

3: Graph Compound Functions

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

y = sin x

3: Graph Compound Functions

x sin x x + sin x0 0 0

/2 1/2 + 1

2.57 0 3.14

3/2 -13/2 - 1

3.712 0 2 6.28

5/2 15/2 + 1

8.85

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

y = cos x

3: Graph Compound Functions

x cos x x + cos x0 1 1

/2 0 /2 1.57 -1 -1 2.14

3/2 03/2 4.71

2 12 +1

7.28

5/2 05/2 7.85

Summary Assignment

Now you know how to graph sinusoidal functions

Ask questions while you finish the assignment

Finish missing workExam

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.

Conclusion