Graphing Sinusoidal Functions

1
Graphing Sinusoidal Functions

• ysin x

2
y sin x

• Recall from the unit circle that

• Using the special triangles and quadrantal
  angles,
• we can complete a table.

3
Table of Values
Quadrant I Quadrant 2
? y





y

.5
.707
.866
1
? y




y
.866
.707
.5

4
Table of Values
Quadrant III Quadrant IV
? y




? y




5
Parent Function ysin x
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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.

• 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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Critical Points

• Between 0 and 2?, there is one maximum point at (
  , 1).

• Between 0 and 2?, there is
• one minimum point at ( , -1).

• Between 0 and 2?, there are three
• zeros at (0,0), (?,0) and (2?,0).

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Parent Function Key Points
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 gt 1, vertical stretch
• If 0ltalt1, vertical shrink
• If a is negative, the graph reflects
• about the x-axis.

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y 3 sin x
What changed?
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y sin x
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y -2 sin x
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ya sin b (x-c)d

• b horizontal stretch or shrink

• Period

• If b gt 1, horizontal shrink
• If 0 lt blt 1, horizontal stretch
• If b lt 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 diving the new period
• by 4.

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y sin 2x
What is the period of this function? If we wanted
to graph only one period, what would the tick
marks need to be?
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y sin x
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y a sin b(x- c ) d

• c horizontal shift
• If c is negative, the graph shifts
• left c units. (xc)(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- )
What changed? Which way did the graph shift? By
how many units?
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y sin (x )
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ya 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.

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y sin x 2
What changed? Which way did the graph shift? By
how many units?
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y sin x - 3
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y 3 sin(2(x-?)) - 2
Can you list all the transformations?
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y-2sin(2x-?) 1