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Understanding the Cotangent Function: Domain, Range, and Asymptotes

This guide explores the cotangent function, focusing on its domain and range, undefined points, and key characteristics. The cotangent, defined as y = cot(x), is undefined at multiples of π, leading to vertical asymptotes. Its domain is all real numbers except kπ, where k is an integer, while its range spans all real numbers. The function's period is π, with properties reflecting vertical and horizontal stretches, shifts, and shrinkage. Graphical representations illustrate these concepts, making understanding cotangent behavior easier.

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Understanding the Cotangent Function: Domain, Range, and Asymptotes

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  1. y = cot x • Recall that • cot  = . • cot  is undefined when y = 0. • y = cot x is undefined at x = 0, x =  and x = 2.

  2. Domain/Range of Cotangent Function • Since the function is undefined at every multiple of , there are asymptotes at these points. • Graphs must contain the dotted asymptote lines. These lines will move if the function contains a horizontal shift, stretch or shrink. • There are asymptotes at every multiple of . • The domain is (-,  except k) • The range of every cot graph is (-, ).

  3. Period of the Function • This means that one complete cycle occurs between zero and . • The period is .

  4. Max and Min Cotangent Function • Range is unlimited; there is no maximum. • Range is unlimited; there is no minimum.

  5. Parent Function Key Points • x = 0: asymptote. The graph approaches  as it approaches this asymptote. • ( , 1) , ( , 0) , ( , -1) • x = : asymptote. The graph approaches - as it approaches this asymptote.

  6. Graph of Parent Functiony = cot x

  7. The Graph: y = a cot 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, the graph reflects about the x-axis.

  8. y = 4 cot x

  9. The Graph: y = a cot b(x-c) +d • b= horizontal stretch or shrink. • Period = . • If |b| > 1, there is a horizontal shrink. • If 0 < |b| < 1, there is a horizontal stretch.

  10. y = cot 2x

  11. The Graph: y = a cot b(x-c) +d • c = horizontal shift. • If c is negative, the graph shifts left c units. • If c is positive, the graph shifts right c units.

  12. y = cot (x - )

  13. The Graph: y = a cot 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.

  14. y = cot x - 4

  15. y = cot (2x + ) + 2

  16. y = - 2cot ( ½ x - ) - 3

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