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

2. Graphs of Sine & Cosine y = sin xy = 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 xy = 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 period = 2  / B It will be assumed that B > 0 since … y = sin (-Bx) = -sin Bx y = cos (-Bx) = cos Bx

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

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

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

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

21. y = A sin (Bx - C) + D y = 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 that B is positive.

22. y = A sin (Bx - C) + D y = 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 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 that B is positive. Always determine the result in this order. {

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