trigonometry, 4.0: students graph functions of the form f(t)=asin(bt+c) or f(t)=acos(bt+c) and...
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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
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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
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Quick check!
Can you find the distance between two numbers?
Can you find the midpoint between two numbers?
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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
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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.
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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
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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.
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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
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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
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State the amplitude, period, phase shift, and vertical shift for the graph of:
2: Write Equations Example
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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
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2: Write Equations Example
YOU TRY! State the amplitude, period, phase shift, and vertical shift for the graph of:
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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)
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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
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Graph y = x + sin x.
3: Graph Compound Functions
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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
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YOU TRY: Graph y = x + cos x.First create a table of each graph: y = x or
y = cos x
3: Graph Compound Functions
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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
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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