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The Sine Graph: Introduction and Transformations
26 April 2011
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The Sine Graph – A Review sin(t) = y t sin(t)
2
23
2
0
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Key Features of y = sin(t)
Maximum:
Minimum:
Domain:
Range:
2
2
– 2
0
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Multiple Revolutionst sin(t)
0
0.5π
1π
1.5π
2π
2.5π
3π
3.5π
4π
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Trigonometric Graphs Repeat!!!
Range: Domain:
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Periodicity Trigonometric graphs
are periodic because
the pattern of the graph
repeats itself
How long it takes the
graph to complete one
full wave or revolution
is called the period
20
2
–21 Period 1 Period
Period: π
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Periodicity, cont.
2
tsiny )t4sin(y
2
2
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Your Turn: Complete problems 1 – 3 on the Identifying
Key Features of Sine Graphs Handout
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Calculating Periodicity If f(t) = sin(bt), then period =
Period is always positive
1. f(t) = sin(–6t) 2.
3.
|b|
2
2
tsin)t(f
4
t3sin)t(f
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Your Turn: Calculate the period of the following graphs:
1. f(t) = sin(3t) 2. f(t) = sin(–4t)
3. 4. f(t) = 4sin(2t)
5. 6.
5
t2sin6)t(f
8
tsin4)t(f
4
tsin)t(f
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Amplitude Amplitude is a trigonometric graph’s greatest
distance from the x-axis. Amplitude is always positive.
If f(t) = a sin(t), then amplitude = | a |
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Calculating Amplitude Examples1. f(t) = 6sin(4t) 2. f(t) = –5sin(6t)
3. 4.)tsin(3
2)t(f
3
tsin
5
1)t(f
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Your Turn: Complete problems 4 – 9 on the Identifying
the Key Features of Sine Graphs handout
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Sketching Sine Graphs – Single Smooth Line!!!
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Transformations Investigation – Investigation #1
2
23
2
t f(t) = sin(t) f(t) = sin(t) + 3
0
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Refection Questions3. What transformations did you see?
4. A.
B.
5. A.
B.
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Transformationsf(t) = a sin(bt – c) + k
Vertical Shift
Pay attention to the parentheses!!!
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Investigation #2!
2
23
2
t f(t) = sin(t) f(t) = 2sin(t)
0
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Reflection Questions4. What transformation did you see?
Stretch = coefficient is a whole # Compression = coefficient is a fraction5. A.
B.C.
6. A.B.C.
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Transformationsf(t) = a sin(bt – c) + k
Vertical Shift
Stretch or Compression
“Amplitude Shift”
Pay attention to the parentheses!!!
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Reflection Questions4. What transformation did you see?
5. A.
B.
C.
6. A.
B.
C.
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Transformationsf(t) = a sin(bt – c) + k
Vertical Shift
Stretch or Compression
“Amplitude Shift”
Pay attention to the parentheses!!!
Period Shift
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Reflection Questions4. What transformation did you see?
4. A.
B.
C.
6. A.
B.
C.
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Transformationsf(t) = a sin(bt – c) + k
Vertical Shift
Stretch or Compression
“Amplitude Shift”
Pay attention to the parentheses!!!
Period ShiftPhase Shift
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Identifying Transformationsf(t) = 2 sin(4t – π) – 3
“Amplitude Shift”:
Period Shift:
Phase Shift:
Vertical Shift:
“Amplitude Shift”:
Period Shift:
Phase Shift:
Vertical Shift:
623
tsin
3
1)t(f
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Your Turn: Identify the transformations of the following
sine graphs:
1. f(t) = 3 sin(t) + 2 2. f(t) = –sin(t – 4) + 1
3. 4.7)t2sin(3
1)t(f 8
5
t2sin
3
2)t(f
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Sketching Transformations Step 1: Identify the correct order of
operations for the function1. Period Shifts
2. Phase Shifts
3. Trig Function
4. “Amplitude Shifts” (Stretches or Compressions)
5. Vertical Shifts
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Sketching Transformations, cont. Step 2: Make a table that follows the order
of operations for the function (Always start with the key points!)
Step 3: Complete the table for the key points (0, , , , )
Step 4: Plot the key points Step 5: Connect the key points with a
smooth line
2 2
3 2
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Example 1: y = –sin(t) + 1
t
0
2
2
3
2
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Example 1: y = –sin(t) + 1
Domain:
Range:
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Example 2: y = 2 sin(t) – 3
t
0
2
2
3
2
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Example 2: y = 2 sin(t) – 3
Domain:
Range:
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Review – Solving for Coterminal Angles If an angle is negative or greater than 2π,
then we add or subtract 2π until the angle is between 0 and 2π. –5π + 2π = –3π + 2π = –π + 2π = π
2
32
2
7
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Your Turn: On a separate sheet of paper (or in the
margin of your notes), find a coterminal angle between 0 and 2π for each of the following angles:
1. 2. 3π 3. 4π
4. 5. 3π
2
2
3
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Problem 6:
t
0
2
2
3
2
2
tsiny
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Problem 6:
Domain:
Range:
2
tsiny
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Problem 7:
t
0
2
2
3
2
tsin6y