Section 5.3a
Graphs of the Cosecant and Secant Functions
RECALL: csc x =
1
so where sin x = 0 , csc x has an asymptote
sin x
Cosecant: f ( x) = csc x
Period: 2 π
Vertical Asymptote: x = kπ , k is an
integer.
x-intercepts: None
y-intercept: None
Domain: x ≠ kπ , k is an
integer.
Range: (−∞,−1] U [1, ∞)
To graph y = A csc( Bx − C ) , first graph, THE HELPER GRAPH, y = A sin( Bx − C ) .
RECALL: sec x =
1
so where cos x = 0 , sec x has an asymptote.
cos x
Secant: f ( x) = sec x
Period: 2 π
Vertical Asymptote: x =
kπ
,k
2
is an odd integer.
x-intercepts: None
y-intercept: (0, 1)
kπ
Domain: x ≠
, k is an odd
2
integer.
Range: (−∞,−1] U [1, ∞)
To graph y = A sec( Bx − C ) , first graph, THE HELPER GRAPH, y = A cos( Bx − C ) .
 πx 
Example 1: Graph f ( x) = − sec  + 1 , but first describe the transformations.
 2 
π

Example 2: Graph f ( x) = 2 csc 2x −  , but first describe the translations.
2

Example 3: Given the equation
these is the graph of the function?
A.
B.
=
+ . First find the period. Which of