1. 6.5 – Translation of Sine and Cosine Functions

2. Phase Shift
A horizontal translation or shift of a trigonometric function
y = Asin(kθ + c) or y = Acos(kθ + c)
The phase shift is -c/k, where k > 0
If c > 0, shifts to the left
If c < 0, shifts to the right

3. State the phase shift for each function. Then graph the function.
y = cos(θ + π)
y = sin(4θ – π)

4. Midline
A horizontal axis that is the reference line or the equilibrium point about which the graph oscillates.
It is in the middle of the maximum and minimum.
y =Asin(kθ + c) + h or y =Acos(kθ + c) + h
The midline is y = h

5. Vertical Shift y =Asin(kθ + c) + h or y =Acos(kθ + c) + h
The vertical shift is h
The midline is y = h
If h < 0, the graph shifts down
If h > 0, the graph shifts up

6. State the vertical shift and the equation for the midline of each function. Then graph the function.
1. y = 3sinθ + 2

7. Putting it all together!
Determine the vertical shift and graph the midline.
Determine the amplitude. Dash lines where min/max values are located.
Determine the period and draw a dashed graph of the sine or cosine curve.
Determine the phase shift and translate your dashed graph to draw the final graph.

8. Graph y = 2cos(θ/4 + π/2) – 1

9. Graph y = -1/2sin(2θ - π) + 3

10. 6.6 – Modeling Real World Data with Sinusoidal Functions
Representing data with a sine function

11. How can you write a sin function given a set of data?
Find the amplitude, “A”: (max – min)/2
Find the vertical translation, “h”: (max + min)/2
Find “k”: Solve 2π/k = Period
Substitute any point to solve for “c”

12. Write a sinusoidal function for the number of daylight hours in Brownsville, Texas.
Month, t 1 2 3 4 5
Hours, h
10.68
11.30
11.98
12.77
13.42 6 7 8 9 10 11 12
13.75
13.60
13.05
12.30
11.57
10.88
10.53

13. 1. Find the amplitude
Max = 13.75
Min = 10.53
(13.75 – 10.53)/2 = 1.61

14. 2. Find “h”
Max = 13.75
Min = 10.53
( )/2 = 12.14

15. 3. Find “k”
Period = 12
2π = 12 k
2π = 12k
π/6 = k

16. 4. Substitute to find “c” y = Asin(kθ – c) + h
y = 1.61sin(π/6 t – c)
10.68 = 1.61sin(π/6 (1) – c)
-1.46 = 1.61sin(π/6 – c)
sin-1(-1.46/1.61) = π/6 – c
1.659… = c

17. 5. Write the function
y = 1.61sin(π/6t – 1.66)

18. Homework Help
Frequency – the number of cycles per unit of time; used in music
Unit of Frequency is hertz
Frequency = 1/Period
Period and Frequency are reciprocals of each other.

19. Assignment Guide Changes
Today’s work:
6.5 p383 #15 – 24 (x3) – now draw the graphs
6.6 p391 #7-8, 13, 22-23
Quiz on Wednesday over 6.3 – 6.5