1. 9) P = π10) P = π/211) P = π/5; ± π/10 12) P = (2 π)/3; ± π/313) P = π/4; ± π/8 14) P = (3π 2 )/2; ±(3π 2 )/4 15) 16) 17) 18)

2. Translating Sine and Cosine Functions

3.  The same translating rules apply to all functions  Each constant in the equation does the same job y = a f (b (x – h)) + k  a – vertical stretch  b – horizontal stretch  h – horizontal shift  k – vertical shift

4.  In periodic functions the horizontal shift is also called the “phase shift”  The phase shift tells us how far around the unit circle we need to start to have the same results What is the value of h in each function? Describe the phase shift in terms of left or right.  g (x) = f (x + 1) h = –1; left 1  m (x) = f (x – 3) h = 3; right 3  y = sin (x + π) h = –π; left π

5. A phase shift moves the graph sideways k moves the graph up or down

6. Translate the graph f (x) to be f (x – 1) 2 4

7.  Parent Functions:  y = a sin bx  y = a cos bx  Translated Functions  y = a sin b (x – h) + k  y = a cos b (x – h) + k  Translating Rules |a| = amplitude = period (x is in radians and b > 0) h = phase shift, or horizontal shift k = vertical shift 2πb2πb

8. For the next class complete #3 – 30 every 3 rd, starting on page 746.

9. 3) h = 1.6; right 1.66) h = 5π/7; right 5π/7 9) 12) 15) 18) Amp: 4, Per: π, Left 1, down 221) 24) 27) 30)

10.  We can use the values for period, amplitude, phase shift, and vertical shift  Begin with the parent function and place the values for a, b, h, and k in their appropriate places Write an equation for each translation: 1) y = sin (x), 4 units down y = sin (x) – 4 2) y = cos (x), π units left y = cos (x + π) 3) y = sin (x), period of 3, amp of 2, right π/2, down 1 y = 2 sin (x – π/2) – 1 2π32π3

11. MonthAverage High January42 February45 March52 April59 May68 June79 July84 August82 September74 October63 November50 December42 Plot a graph of the data (in degrees) and write a cosine function to model the information. Let a > 0. J J JFM MAASOND 50˚F J y = 21 cos (x – 180) + 63

12. For the next class complete #34 – 43 starting on page 746.