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
Sine


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The most fundamental sine wave, y=sin(x),
has the graph shown.
It fluctuates from 0 to a high of 1, down to –1, and
back to 0, in a space of 2.
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
The graph of y  a sin b( x  h)  k is determined by
four numbers, a, b, h, and k.
The amplitude, a, tells the height of each peak and
the depth of each trough.
 The frequency, b, tells the number of full wave
patterns that are completed in a space of 2.
 The period of the function is
 The two remaining numbers, h and k, tell the
2
translation of the wave from the origin.

b
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5
4

3
2
1
2
1
1
1
2


3

4

5
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2
Which of the following
equations best describes
the graph shown?

(A) y = 3sin(2x) - 1
(B) y = 2sin(4x)
(C) y = 2sin(2x) - 1
(D) y = 4sin(2x) - 1
(E) y = 3sin(4x)
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5
4

3
2

1
2
1
1
1
2
2

4
5
y = 3sin(2x) - 1

Graph is translated -1
vertically.
Find height of each peak.

3
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Find the baseline between
the high and low points.
Amplitude is 3
Count number of waves
in 2

Frequency is 2
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
Cosine


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The graph of y=cos(x) resembles the graph of
y=sin(x) but is shifted, or translated, units to
the left.
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It fluctuates from 1
to 0, down to –1,
back to 0 and up to
1, in a space of 2.
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
The values of a, b, h, and k change the shape
and location of the wave as for the sine.
y  a cos b( x  h)  k
Amplitude
Frequency
Period
Translation
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a
b
2/b
h, k
Height of each peak
Number of full wave patterns
Space required to complete wave
Horizontal and vertical shift
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
Which of the following
equations best describes
the graph?





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(A) y = 3cos(5x) + 4
(B) y = 3cos(4x) + 5
(C) y = 4cos(3x) + 5
(D) y = 5cos(3x) +4
(E) y = 5sin(4x) +3
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6
4
2
2
1
1
2
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
Find the baseline


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4
2
Amplitude = 5
Number of waves in
2

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Vertical translation + 4
Find the height of
peak


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Frequency =3
2
1
1
2
y = 5cos(3x) + 4
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
Tangent


The tangent function has a
discontinuous graph,
repeating in a period of .
Cotangent

Like the tangent, cotangent is
discontinuous.
 Discontinuities of the
cotangent are
units left of
those for tangent.

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
Secant and Cosecant


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The secant and cosecant functions are the
reciprocals of the cosine and sine functions
respectively.
Imagine each graph is balancing on the peaks and
troughs of its reciprocal function.
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