graphing sinusoidal functions y=sin x. recall from the unit circle that: –using the special...
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![Page 1: Graphing Sinusoidal Functions y=sin x. Recall from the unit circle that: –Using the special triangles and quadrantal angles, we can complete a table](https://reader035.vdocuments.us/reader035/viewer/2022062516/56649e3f5503460f94b2fb88/html5/thumbnails/1.jpg)
Graphing Sinusoidal Functions
y=sin x
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y= sin x
•Recall from the unit circle that:
sin .yr
–Using the special triangles and quadrantal angles, we can complete a table.
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y
6
1
2
4
1
2
3
3
2
2
Quadrant I Quadrant 2
0
1
0
y .5 .707 .866 1
0
y
2
3
3
2
y .866 .707 .5
1
23
4
5
6
1
2
00
Table of Values
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y
Quadrant III Quadrant IV
5
4
4
3
7
6
3
2
1
2
1.707
2
3.866
2
1
Table of Values
y5
3
7
4
11
6
1.5
2
2 0
3.866
2
1.707
2
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3
6
4
2
5
6
4
3
7
6
5
4
4
3
3
2
5
3
7
4
3
4
11
6
2
Parent Functiony=sin x
0
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Domain•Recall that we can rotate
around the circle in either direction an infinite number of times.•Thus, the domain is (- , )
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Range•Recall that –1 sin 1.
1
1
•Thus the range of this function is [-1 , 1 ]
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Period
•One complete cycle occurs between 0 and 2.
•The period is 2.
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How many periods are shown?
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•Between 0 and 2, there is one minimum point at ( , -1).
Critical Points
•Between 0 and 2, there is one maximum point at ( , 1).
2
3
2
•Between 0 and 2, there are three zeros at (0,0), (,0) and (2,0).
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Parent FunctionKey Points
2
3
2
2
0
1
1
* Notice that the key points of the graph separate the graph into 4 parts.
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y= a sin b(x-c)+d
•a = amplitude, the distance from the center to the maximum or minimum.
• If |a| > 1, vertical stretch • If 0<|a|<1, vertical shrink • If a is negative, the graph
reflects about the x-axis.
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y = 3 sin x
2
3
2
20
1
1
What changed?
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y= sin x1
4
1
1
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y= -2 sin x
1
1
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y=a sin b (x-c)+d
•b= horizontal stretch or shrink2b
• Period =
•If |b| > 1, horizontal shrink •If 0 < |b|< 1, horizontal stretch•If b < 0, the graph reflects about the y-axis.
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Tick Marks
•Recall that the key points separate the graph into 4 parts.•If we alter the period, we need to alter the x-scale.•This can be done by dividing the new period by 4.
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y = sin 2x
3
2
20
2
1
1
What is theperiod ofthis function?If we wanted to graph only one period, what would the tick marks need to be?
4
3
4
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y = sin x 1
3
1
1
2
3
2
2
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y = a sin b(x- c ) + d
• c = horizontal shift• If c is negative, the graph shifts left c units. (x+c)=(x-(-c))• If c is positive, the graph shifts right c units. (x-c)=(x-(+c))• In trigonometric functions, these horizontal shifts are called phase shifts.
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y = sin(x- )2
1
0
12
3
2
2
What changed?Which way did the graph shift?By how many units?
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y = sin (x + )
2
3
2
2
1
1
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y=a sin b(x-c) + d
• d= vertical shift• If d is positive, graph shifts up d units.• If d is negative, graph shifts down d units.• In trigonometric functions, these vertical shifts are called the vertical displacement.
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y = sin x +2
1
0
12
3
2
2
What changed?
Which way did the graph shift?
By how many units?
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y = sin x - 31
12
3
2
2
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y = 3 sin(2(x-)) - 2
1
0
1 2
3
2
2
Can you list all thetransformations?
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y=-2sin(2x-) +1
23
2
2
1
1