Math 2204/05
Graph Æ Equation
Trigonometric Functions
Chapter 3
Steps+
1. Determine the following by viewing the graph.
• Period / Amplitude/ Sinusoidal Axis
2. Determine the following from the information found in step i.
•
Horizontal Stretch (HS) = image period = image period
original period
3600
• Vertical Stretch (VS) = Amplitude
• Vertical Translation (VT) Æ y = (VT) (from Sinusoidal Axis)
• Horizontal Translation (HT)
a. Find intersection of y-intercept and sinusoidal axis
b. Go right until you hit the graph at a point where the graph starts
to turn upwards.
c. Determine the horizontal distance from the intersection point to
the point where you stopped.
3. Write the mapping rule.
• (x, y) Æ [(HS)x ± (HT), (VS)y ± (VT)]
4. Write the equation in terms of sin.
0
0
• Remember: sin(x) and cos(x) are out of phase 90 [i.e. (1/4)(360 )]
• Therefore, to write in terms of cos change the (HT of sin) as follows.
(HT of Cos) = (HT of Sin) + [(1/4)(image period)]
Examples
1.
Period
Sinusoidal Axis
Amplitude
Math 2204/05
Graphing Transformations
New Start Point
1
Period
Amplitude
Sinusoidal Axis
Start Point
Mapping Rules
4050 - 450 = 3600
3
Y = -2
(3150, -2)
SIN
HT = 3150 + (1/4)(3600)
COS
= 3150 + (900)
= 4050
Equations
SIN
COS
HS = 360/360 = 1
VS = 3
VT = -2
HT = 3150
(x, y) Æ (x + 3150, 3y - 2)
(x, y) Æ (x + 4050, 3y - 2)
1/3(y + 2) = sin(x - 3150)
1/3(y + 2) = cos(x - 4050)
2.
Period
Amplitude
Sinusoidal Axis
Start Point
Mapping Rules
2100 - 300 = 1800
4
Y=1
(750, 1)
SIN
HT = 750 + (1/4)(1800)
COS
= 750 + (450)
=1200
Equations
Math 2204/05
Graphing Transformations
SIN
COS
HS = 180/360 = 1/2
VS = 4
VT = 1
HT = 750
(x, y) Æ (1/2x + 750, 4y + 1)
(x, y) Æ (1/2x + 1200, 4y + 1)
1/4(y - 1) = sin2(x - 750)
1/4(y - 1) = cos2(x - 1200)
2
3.
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6.
Math 2204/05
Graphing Transformations
3
7.
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10.
Math 2204/05
Graphing Transformations
4