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y = csc X. Recall from the unit circle that csc = r/y. csc is undefined whenever y = 0. y = csc x is undefined at x = 0 , x = and x= 2 . Domain of Cosecant Function.
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y = csc X • Recall from the unit circle that csc = r/y. • csc is undefined whenever y = 0. • y = csc x is undefined at x = 0, x = and x=2.
Domain of Cosecant Function • Since the cosecant function is undefined at multiples of , there is an asymptote at those points. They will move if the function contains a horizontal shift, stretch or shrink. • The domain is (-, except k)
Period of y = csc x • One complete cycle occurs between 0 and 2. • The period is 2.
Range of y = csc x • The range of every csc graph varies depending on vertical shifts • The range of the parent graph is • (-,-1][1, )
Max and Min of y = csc x • There is a local max at (3/2, -1) • There is a local min at (/2, 1)
Parent Function y = cscX x = 0: asymptote. The graph approaches as it nears this asymptote. • x = : asymptote. The graph approaches as it nears this asymptote.
Parent Function y = cscX • x = 0: asymptote. The graph approaches - as it nears this asymptote. • x = 2: asymptote. The graph approaches - as it nears this asymptote.
Graph of Parent Function • Recall that the cosecant function is the • reciprocal of the sine function.
The Graph: y = a csc b(x-c) + d • a = vertical stretch or shrink • If |a| > 1, there is a vertical stretch. • If 0 < |a| < 1, there is a vertical shrink. • If a is negative, there is a reflection about the x-axis.
The Graph: y = a csc b (x-c) + d • b= horizontal stretch or shrink • Period = 2/b • If |b| > 1, horizontal shrink • If 0 < |b| < 1, horizontal stretch
The Graph: y = a csc b (x-c) + d • c = horizontal shift • If c is negative, the graph shifts left c units. (x-(-c)) = (x+c) • If c is positive, the graph shifts right c units. (x-(+c)) = (x-c)
The Graph: y = a csc b(x-c) + d • d = vertical shift • If d is positive, the graph shifts up d units. • If d is negative, the graph shifts down d units.
