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Graphing Tangent and Cotangent

This guide provides a detailed overview of graphing the tangent and cotangent curves, focusing on key points and periodicity. Learn how to effectively use a graphing calculator to visualize y = tan(x) and y = cot(x) across different intervals, utilizing radians for accuracy. Clear examples highlight transformations including vertical shifts and horizontal phase shifts. Understand key concepts such as periods, amplitudes, and intercepts for these trigonometric functions, enabling better comprehension of their behaviors and applications in mathematics.

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Graphing Tangent and Cotangent

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  1. Graphing Tangent and Cotangent

  2. Graphing Tangent Curve 0 /6 /4 /3 /2 2/3 3/4 5/6  7/6 5/4 4/3 3/2 5/3 7/4 11/6 2 • y = tan (x)

  3. Graphing Cotangent Curve 0 /6 /4 /3 /2 2/3 3/4 5/6  7/6 5/4 4/3 3/2 5/3 7/4 11/6 2 • y = cot (x)

  4. Using the Graphing Calculator • Mode— • Radians • Function • Sequential • Window— • X min = -  • X max = 3 • X scale = /6 • Window— • Y min =-3 • Y max = 3 • Y scale = .5

  5. Press Y= • y1 = tan (x) • y2 = 3tan (x) Press Graph

  6. Press Y= • y1 = tan (x) • y2 = 4tan (x) Press Graph

  7. Press Y= • y1 = tan (x) • y2 = -4tan (x) Press Graph

  8. Press Y= • y1 = tan (x) • y2 = tan (x) +1 Press Graph

  9. Press Y= • y1 = tan (x) • y2 = tan (x +1) Press Graph

  10. Press Y= • y1 = tan (x) • y2 = tan (2x) Press Graph

  11. Press Y= • y1 = tan (x) • y2 = tan (1/2x) Press Graph

  12. Vertical Shift up/down Y = A tan (Bx - C) ± D Period  |B| No Amplitude, but determines two points on the graph. Horizontal Shift Left/Right C B

  13. y = A tan (Bx – C) + D Intercept Maximum? Asymptote Minimum? Intercept Unless A = Negative # Intercept Minimum? Asymptotes Maximum? Intercept

  14. y = cot (x) Press Y= • y1 = 1/tan (x) • y2 = 3/tan (x) Press Graph

  15. Press Y= • y1 = 1/tan (x) • y2 = 4/tan (x) Press Graph

  16. Press Y= • y1 = 1/tan (x) • y2 = -4/tan (x) Press Graph

  17. Press Y= • y1 = 1/tan (x) • y2 = 1/tan (x) +1 Press Graph

  18. Press Y= • y1 = 1/tan (x) • y2 = 1/tan (x +1) Press Graph

  19. Press Y= • y1 = 1/tan (x) • y2 = 1/tan (2x) Press Graph

  20. Vertical Shift up/down Y = A cot (Bx - C) ± D Period  |B| No Amplitude, but determines two points on the graph. Horizontal Shift Left/Right C B

  21. y = A cot (Bx – C) + D Asymptote Maximum? Intercept Minimum? Asymptote Unless A = Negative # Asymptote Minimum? Intercept Maximum? Asymptote

  22. Examples-- • y = 3tan (x + 1) -3 Amp.= _______ Period = _____ Horizontal Phase Shift = _________ Vertical Phase Shift = ________ • y = cot (x - 2) Amp.= _______ Period = _____ Horizontal Phase Shift = _________ Vertical Phase Shift = ________

  23. y = cot (2x) Amp.= _______ Period = _____ Horizontal Phase Shift = _________ Vertical Phase Shift = ________ y = ½ tan (½ x) + 1 Amp.= _______ Period = _____ Horizontal Phase Shift = _________ Vertical Phase Shift = ________

  24. Graphing by Key Points • y = 2 tan x Amp = _________ Horizontal Shift = _______ Period = _______ Vertical Shift = _________ Inc. = ________ (0, ) (/4, ) (/2, ) (3/4, ) (, )

  25. Graphing by Key Points • y = -2 cot x Amp = _________ Horizontal Shift = _______ Period = _______ Vertical Shift = _________ Inc. = ________ (0, ) (/4, ) (/2, ) (3/4, ) (, )

  26. Graphing by Key Points • y = tan (1/2x) Amp = _________ Horizontal Shift = _______ Period = _______ Vertical Shift = _________ Inc. = ________ (0, ) (/2, ) ( , ) (3/2, ) (2, )

  27. Graphing by Key Points • y = 3 cot 1/2x Amp = _________ Horizontal Shift = _______ Period = _______ Vertical Shift = _________ Inc. = ________ (0, ) (/2, ) (, ) (3/2, ) (2, )

  28. Graphing by Key Points • y = 3 tan x +2 Amp = _________ Horizontal Shift = _______ Period = _______ Vertical Shift = _________ Inc. = ________ (0, ) (/4 ) (/2, ) (3/4, ) (, )

  29. Graphing by Key Points • y = cot x - 2 Amp = _________ Horizontal Shift = _______ Period = _______ Vertical Shift = _________ Inc. = ________ (0, ) (/4 ) (/2, ) (3/2, ) (, )

  30. Graphing by Key Points • y = -tan (x + ) Amp = _________ Horizontal Shift = _______ Period = _______ Vertical Shift = _________ Inc. = ________ (-, ) (-3/4 , ) (-/2 , ) (-/4, ) (0, )

  31. Graphing by Key Points • y = cot (x - /4) Amp = _________ Horizontal Shift = _______ Period = _______ Vertical Shift = _________ Inc. = ________ (/4 , ) (/2, ) (3/4, ) (, ) (5/4, )

  32. Graphing by Key Points • y = 4tan (x/2 + /2) Amp = _________ Horizontal Shift = _______ Period = _______ Vertical Shift = _________ Inc. = ________ (-, ) (-/2 , ) (0, ) (/2, ) (, )

  33. Graphing by Key Points • y = -cot x Amp = _________ Horizontal Shift = _______ Period = _______ Vertical Shift = _________ Inc. = ________ (0, ) (1/4, ) (1/2, ) (3/4, ) (1, )

  34. Graphing by Key Points • y = 2tan (x/2 - /2) -1 Amp = _________ Horizontal Shift = _______ Period = _______ Vertical Shift = _________ Inc. = ________ (, ) (3/2, ) (2, ) (5/2, ) (3, )

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