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MTH 112 Elementary Functions Chapter 5 The Trigonometric Functions

MTH 112 Elementary Functions Chapter 5 The Trigonometric Functions. Section 6 – Graphs of Transformed Sine and Cosine Functions. Graphs of Sine & Cosine. y = sin x. y = cos x. Period = 2  Amplitude = 1.

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MTH 112 Elementary Functions Chapter 5 The Trigonometric Functions

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  1. MTH 112Elementary FunctionsChapter 5The Trigonometric Functions Section 6 – Graphs of Transformed Sine and Cosine Functions

  2. Graphs of Sine & Cosine y = sin x y = cos x Period = 2 Amplitude = 1 What will different constants in different locations do to these functions?

  3. Review of Transformations of Functions • Compare: y = f(x) and y = -f(x) • Reflection wrt the x-axis.

  4. Review of Transformations of Functions • Compare: y = f(x) and y = f(-x) • Reflection wrt the y-axis

  5. Review of Transformations of Functions • Compare: y = f(x) and y = a f(x), a > 1 • Vertical Stretch

  6. Review of Transformations of Functions • Compare: y = f(x) and y = a f(x), 0 < a < 1 • Vertical Compression

  7. Review of Transformations of Functions • Compare: y = f(x) and y = f(bx), b > 1 • Horizontal Compression

  8. Review of Transformations of Functions • Compare: y = f(x) and y = f(bx), 0 < b < 1 • Horizontal Stretch

  9. Review of Transformations of Functions • Compare: y = f(x) and y = f(x - c), c > 0 • Horizontal Shift Right

  10. Review of Transformations of Functions • Compare: y = f(x) and y = f(x - c), c < 0 • Horizontal Shift Left

  11. Review of Transformations of Functions • Compare: y = f(x) and y = f(x) + d, d > 0 • Vertical Shift Up

  12. Review of Transformations of Functions • Compare: y = f(x) and y = f(x) + d, d < 0 • Vertical Shift Down

  13. Graphs of Sine & Cosine y = sin x y = cos x Period = 2 Amplitude = 1 What will different constants in different locations do to these functions?

  14. y = A sin x Amplitude = |A| A < 0 reflex wrt x-axis.

  15. y = sin Bx It will be assumed that B > 0 since … y = sin (-Bx) = -sin Bx y = cos (-Bx) = cos Bx period = 2 / B

  16. y = sin (x - C) C > 0  phase shift right C < 0  phase shift left

  17. y = sin (Bx - C) As before, assume that B > 0. Otherwise modify the equation. Could rewrite as … y = sin [B(x – C/B)] period = 2/B phase shift  C/B left (+) or right(-)

  18. y = sin x + D D > 0  translate up D < 0  translate down

  19. y = A sin (Bx - C) + Dy = A cos (Bx - C) + D Everything in the previous slides applies in the same way to y = cos x.

  20. y = A sin (Bx - C) + Dy = A cos (Bx - C) + D cos x is just a phase shift of sin x

  21. y = A sin (Bx - C) + Dy = A cos (Bx - C) + D • Any number of these constants can be included resulting in a combination of results. They may … • stretch • compress • reflect • phase shift • translate Which constants affect which characteristics? It can be assumed thatB is positive.

  22. { Always determine the result in this order. y = A sin (Bx - C) + Dy = A cos (Bx - C) + D • Any number of these constants can be included resulting in a combination of results. They may … • stretch  |A| > 1 & B < 1 • compress  |A| < 1 & B > 1 • reflect  wrt the x-axis: A < 0 • phase shift  C/B right (+) or left (-) • translate  D up (+) or down (-) Amplitude = |A| Period = 2 / B It can be assumed thatB is positive.

  23. One more problem … • What would the graph of y = x + sin x look like?

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