1 / 23

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.

gautam
Download Presentation

MTH 112 Elementary Functions Chapter 5 The Trigonometric Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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?

More Related