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(Section 4.6: Graphs of Other Trig Functions)
4.63
Example
Use the Frame Method to graph one cycle of the graph of
y
=
2tan
2
5
x
⎛
⎝
⎜
⎞
⎠
⎟
−
3
. (There are infinitely many possible cycles.)
Solution
Fortunately,
b
=
2
5
>
0
. If
b
<
0
, we would need to use the Even/Odd
Properties. Remember that both tan and cot are odd functions.
P
ivot:
p
=
0,
d
=
−
3
( )
C
ycle shape: We have a tan graph with
a
=
2
>
0
, so we will use:
“
A
mplitude”
=
a
=
2
=
2
P
eriod
=
π
b
=
2/5
=
5
2
I
ncrement
=
1
4
Period
( )
=
1
4
5
2
⎛
⎝
⎜
⎞
⎠
⎟
=
5
8
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4.64
The Frame
Since there was no discernible phase shift, we see some nice symmetry
between the positive and negative
x
coordinates on this tan frame.
Note: If you would prefer to start the labeling process at the “leftcenter”
point, just as for sin and cos cycles, you could find the
x
coordinate of a VA
by setting the argument of tan equal to
−
π
2
or
2
(for example), which
correspond to asymptotes for
y
=
tan
x
, and solving for
x
. Here:
2
5
x
=
−
2
x
=
5
2
⎛
⎝
⎜
⎞
⎠
⎟
−
2
⎛
⎝
⎜
⎞
⎠
⎟
x
=
−
5
4
If you are dealing with a cot graph, then you would set the argument
of cot equal to 0, just as for sin and cos graphs.
(Section 4.6: Graphs of Other Trig Functions)
4.65
PART G: GRAPHS OF CSC AND SEC FUNCTIONS (“UPU, DOWNU” GRAPHS)
Remember that
csc
x
=
1
sin
x
.
How can we use the graph of
y
=
sin
x
to obtain the graph of
y
=
csc
x
?
1)
Draw VAs through the
x
intercepts (in purple) of the
sin
x
graph.
2)
Between any consecutive pair of VAs:
If the
sin
x
graph lies
above
the
x
axis, then draw an
“upU”
that has as its
minimum
point (in brown) the
maximum
point of the sin graph and that
approaches both VAs.
If the
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.
 Fall '11
 staff
 Calculus

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