6 5 Translation of Sine and Cosine Functions

6. 5 – Translation of Sine and Cosine Functions

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

State the phase shift for each function. Then graph the function. 1. y = cos(θ + π) 2. y = sin(4θ – π)

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

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

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

Putting it all together! 1. Determine the vertical shift and graph the midline. 2. Determine the amplitude. Dash lines where min/max values are located. 3. Determine the period and draw a dashed graph of the sine or cosine curve. 4. Determine the phase shift and translate your dashed graph to draw the final graph.

Graph y = 2 cos(θ/4 + π/2) – 1

Graph y = -1/2 sin(2θ - π) + 3

6. 6 – Modeling Real World Data with Sinusoidal Functions • Representing data with a sine function

How can you write a sin function given a set of data? 1. 2. 3. 4. 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”

Write a sinusoidal function for the number of daylight hours in Brownsville, Texas. Month, t Hours, h 1 2 3 4 5 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

1. Find the amplitude Max = 13. 75 Min = 10. 53 (13. 75 – 10. 53)/2 = 1. 61

2. Find “h” Max = 13. 75 Min = 10. 53 (13. 75 + 10. 53)/2 = 12. 14

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

4. Substitute to find “c” y = Asin(kθ – c) + h y = 1. 61 sin(π/6 t – c) + 12. 14 10. 68 = 1. 61 sin(π/6 (1) – c) + 12. 14 -1. 46 = 1. 61 sin(π/6 – c) sin-1(-1. 46/1. 61) = π/6 – c 1. 659… = c

5. Write the function y = 1. 61 sin(π/6 t – 1. 66) + 12. 14

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.

Assignment Guide Changes • Today’s work: 6. 5 p 383 #15 – 24 (x 3) – now draw the graphs 6. 6 p 391 #7 -8, 13, 22 -23 • Quiz on Wednesday over 6. 3 – 6. 5