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Cosecant

Cosecant. Objective. To graph the cosecant. 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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Cosecant

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  1. Cosecant

  2. Objective • To graph the cosecant

  3. 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.

  4. 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)

  5. Range of Cosecant Function • The range of every cosecant graph varies depending on vertical shifts. • The range of the parent graph is (-, -1]U[1, )

  6. Period of y = csc x • One complete cycle occurs between 0 and 2. • The period is 2.

  7. Max and Min of y = csc x • There is a local max at (3/2, -1) • There is a local min at (/2, 1)

  8. Parent Function y = csc x • x = 0: asymptote • X = π • x = 2: asymptote.

  9. Graph of Parent Function • Recall that the cosecant function is the • reciprocal of the sine function.

  10. 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.

  11. y = 2 csc x

  12. To find the asymptotes • Set b(x – c) = 0 • Set b(x – c) = π • Set b(x – c) = 2π

  13. 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

  14. Y = csc ½ x

  15. 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)

  16. Y = csc (x- π/2 )

  17. 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.

  18. Y = csc x - 2

  19. Y = -2 csc ( ½ x + ) - 3

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