Graphs of Secant and
Cosecant
Plan for the Day
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Review Homework
Secant and Cosecant
Homework
Next time: Quiz on Graphing Sine and Cosine
(including bx – c)
2
Get out your graphing calculator…
Graph the following
y = cos x
y = sec x
What do you see??
3
Graph of the Secant Function
1
sec
x
=
The graph y = sec x, use the identity
cos x
.
At values of x for which cos x = 0, the secant function is undefined
and its graph has vertical asymptotes.
y
y = sec x
Properties of y = sec x
4
1. domain : all real x
π
x ≠ kπ + (k ∈ Ζ)
y = cos x
2
2. range: (–∞,–1] ∪ [1, +∞)
3. period: 2π
4. vertical asymptotes:
π
x = kπ + (k ∈ Ζ )
2
x
−
π
π
2
2
π
3π
2
2π
5π
2
3π
−4
5
First graph:
• y = 2cos (2x – π) + 1
Then try:
• y = 2sec (2x – π) + 1
6
Graph
Graph the following
y = sin x
y = csc x
What do you see??
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Graph of the Cosecant Function
1
To graph y = csc x, use the identity csc x =
.
sin x
At values of x for which sin x = 0, the cosecant function
is undefined and its graph has vertical asymptotes.
y
Properties of y = csc x
4
y = csc x
1. domain : all real x
x ≠ kπ (k ∈ Ζ )
2. range: (–∞,–1] ∪ [1, +∞)
3. period: 2π
4. vertical asymptotes:
x = kπ (k ∈ Ζ )
where sine is zero.
x
−
π
π
2
2
π
3π
2
2π
5π
2
y = sin x
−4
9
First graph:
• y = -3 sin (½x + π/2) – 1
Then try:
• y = -3 csc (½x + π/2) – 1
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Key Steps in Graphing Secant
and Cosecant
Identify the key points of your reciprocal graph (sine/cosine),
note the original zeros, maximums and minimums
Find the new period (2π/b)
Find the new beginning (bx - c = 0)
Find the new end (bx - c = 2π)
Find the new interval (new period / 4) to divide the new
reference period into 4 equal parts to create new x values for
the key points
Adjust the y values of the key points by applying the change in
height (a) and the vertical shift (d)
Using the original zeros, draw asymptotes, maximums become
minimums, minimums become maximums…
Graph key points and connect the dots based upon known
shape
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