Adv Precalculus
Worksheet: Section 4.6B
Name: _______________________________
Classwork for Tuesday, Nov 25, 2014
Graphing Cosecant and Secant Functions
Consider the graph of the (parent) COSECANT function:
The PERIOD of the cosecant function is 2𝜋
The DOMAIN is all real numbers (as angle measures) EXCEPT 𝑥 = 𝑛𝜋 , where n is an integer
(this means there are vertical asymptotes!)
The RANGE is (−∞, −1] ∪ [1 , ∞)
VERTICAL ASYMPTOTES are 𝑥 = 𝑛𝜋 , where n is an integer.
SYMMETRY: Origin
Consider the graph of the (parent) SECANT function:
The PERIOD of the tangent function is 2𝜋
𝜋
The DOMAIN is all real numbers (as angle measures) EXCEPT 𝑥 = 2 + 𝑛𝜋 , where n is an integer
(this means there are vertical asymptotes!)
The RANGE is (−∞, −1] ∪ [1 , ∞)
𝜋
VERTICAL ASYMPTOTES are 𝑥 = 2 + 𝑛𝜋 , where n is an integer.
SYMMETRY: y-axis
Transformations to the cosecant and secant functions follow the same pattern as the
transformations to all other functions .
𝑦 = 𝑎 csc(𝑐 𝜃 − ℎ) + 𝑘
and
𝑦 = 𝑎 sec(𝑐 𝜃 − ℎ) + 𝑘
a – represents AMPLITUDE (aka a vertical stretch if a > 1 and a vertical compression if 0 < a < 1)
causes a reflection over the x-axis if a < 0
c – contributes to changes in period, causing a horizontal stretch if 0 < c < 1, and a horizontal
compression if c > 1
2𝜋
Formula for period of cosecant and secant functions: 𝑝𝑒𝑟𝑖𝑜𝑑 =
𝑐
Note: it’s very helpful to draw the “backbone” reciprocal function, then the vertical asymptotes
that set up the period (or transformed period, including phase shifts) first and then fill in the rest
of the graph.
h – in combination with c, contributes to phase shifts, causing a horizontal shift left if h < 0, and a
horizontal shift right if h > 0. Note: it’s helpful to draw in the vertical asymptotes
Formula for phase shift is: 𝑝ℎ𝑎𝑠𝑒 𝑠ℎ𝑖𝑓𝑡 =
ℎ
𝑐
Note: it’s very helpful to draw the “backbone” reciprocal function, then the vertical asymptotes
that set up the period (or transformed period, including phase shifts) first and then fill in the rest
of the graph.
k – contributes to a vertical shift of the MIDLINE (and the rest of the graph) up if k > 0, and down
if k < 0.
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Homework examples: Graph ON GRAPH PAPER, the following equations. Show all detail that
helps you draw the graph accurately (label the x-axis with angle measures, label the y-axis with
appropriate values, sketch in any vertical asymptotes that help guide the shape of the graph).
1) 𝑦 = 3 csc
𝑥
Draw the graph over the interval [−4𝜋, 4𝜋]
2
2) 𝑦 = −2 sec(4𝑥) + 2
Draw the graph over the interval [−𝜋, 𝜋]
3) 𝑦 = csc(2𝑥 − 𝜋)
Draw the graph over the interval [−𝜋, 𝜋]
1
𝜋𝑥
3
2
4) 𝑦 = sec (
𝜋
+ )
2
Draw the graph over the interval [−
3𝜋 3𝜋
2
,
2
]