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Graphing Trigonometric Functions

Graphing Trigonometric Functions. Chapter 4. The sine and cosine curves. Graph y = sinx. The sine and cosine curves. Graph y = cosx. The sine and cosine curves. Graph y = - cosx. The sine and cosine curves. Graph y = - sinx. Amplitude “a”. y = asinx y = acosx

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Graphing Trigonometric Functions

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  1. Graphing Trigonometric Functions Chapter 4

  2. The sine and cosine curves • Graph y = sinx

  3. The sine and cosine curves • Graph y = cosx

  4. The sine and cosine curves • Graph y = -cosx

  5. The sine and cosine curves • Graph y = -sinx

  6. Amplitude “a” y = asinx y = acosx The amplitude will stretch the graph vertically. The value of “a” is half the distance of the max and min.

  7. Amplitude “a” • Graph y = 3cosx

  8. Period of the sine and cosine y = sinbx and y = cosbx The period of the function will shrink or stretch the graph horizontally. The period of a function is The standard period is 2π, this occurs when b = 1.

  9. Period of the sine and cosine • Graph y = sin3x

  10. Period of the sine and cosine • Graph y = cos2x

  11. Amplitude “a” and Period ”b” • Graph y = 3sin4x

  12. Amplitude “a” and Period ”b” • Graph y = -4cosπx

  13. Phase Shifts of sine and cosine y = sinb(x-d) and y = cosb(x-d) The period of the function will have new endpoints when solving the inequality 0 ≤ b(x-d) ≤ 2π. (x – d) is a shift of “d” to the right (x + d) is a shift of “d” to the left

  14. Phase Shifts of sine and cosine • Graph

  15. Phase Shifts of sine and cosine • Graph

  16. Vertical Translations of sine and cosine y = c + sinx and y = c + cosx The “c” will shift the entire graph “c” units up when “c” is positive and “c” units down when “c” is negative

  17. Vertical Translations of sine and cosine • Graph y = 2 + sinx

  18. Vertical Translations of sine and cosine • Graph y = -2 + cos3x

  19. Combinations of Translations • Graph y = -2 – 2sin5x

  20. Combinations of Translations • Graph y = 1 -2cos3(x+π)

  21. Combinations of Translations • Graph

  22. Identifying Features Give the amplitude, period, phase shift, and vertical translation. Amplitude: 2 Period: 2π Phase Shift: π/3 to the left Vertical Translation: none

  23. Identifying Features Give the amplitude, period, phase shift, and vertical translation. Amplitude: 1 Period: 2π/3 Phase Shift: π/6 to the right Vertical Translation: up 1

  24. Identifying Features Give the amplitude, period, phase shift, and vertical translation. Amplitude: 4 Period: π Phase Shift: π to the right Vertical Translation: down 2

  25. Graphs of Secant and Cosecant • Graph y = secx

  26. Graphs of Secant and Cosecant • Graph y = cscx

  27. Graphs of Secant and Cosecant • Graph y = 2csc5x

  28. Graphs of Secant and Cosecant Find the amplitude, period, phase shift, and vertical translation…then graph it. Amplitude: not applicable Period: π Phase Shift: π/6 to the left Vertical Translation: down 1

  29. Graphs of Secant and Cosecant Find the amplitude, period, phase shift, and vertical translation…then graph it.

  30. Graphs of Secant and Cosecant Find the amplitude, period, phase shift, and vertical translation…then graph it. Amplitude: not applicable Period: 2π Phase Shift: π/4 to the right Vertical Translation: up 2

  31. Graphs of Secant and Cosecant Find the amplitude, period, phase shift, and vertical translation…then graph it.

  32. Over “2-periods” • Graph y = sinx

  33. Over “2-periods” • Graph

  34. Tangent and Cotangent • Sine,Cosine,Secant, and Cosecant have a standard period of 2π. • The tangent and cotangent have a standard period of π. • The standard tangent graph has asymptotes at –π/2 and π/2 • The standard cotangent graph has asymptotes at 0 and π

  35. Tangent and Cotangent • Graph y = tanx

  36. Tangent and Cotangent • Graph y = cotx

  37. Tangent and Cotangent • Graph y = 1 – tan3x

  38. Tangent and Cotangent • Graph y = 2 + 3cot(x – π) Find the amplitude, period, phase shift, and vertical translation…then graph it. Amplitude: not applicable Period: π Phase Shift: π to the right Vertical Translation: up 2

  39. Tangent and Cotangent • Graph y = 2 + 3cot(x – π) Find the amplitude, period, phase shift, and vertical translation…then graph it.

  40. Tangent and Cotangent • Graph Find the amplitude, period, phase shift, and vertical translation…then graph it. Amplitude: not applicable Period: π/2 Phase Shift: π/8 to the left Vertical Translation: up 1

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