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. y = cos(θ + π) 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 = 3sinθ + 2

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.

Graph y = 2cos(θ/4 + π/2) – 1

Graph y = -1/2sin(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? 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 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

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 = 12k π/6 = k

4. Substitute to find “c” y = Asin(kθ – c) + h y = 1.61sin(π/6 t – c) + 12.14 10.68 = 1.61sin(π/6 (1) – c) + 12.14 -1.46 = 1.61sin(π/6 – c) sin-1(-1.46/1.61) = π/6 – c 1.659… = c

5. Write the function y = 1.61sin(π/6t – 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 p383 #15 – 24 (x3) – now draw the graphs 6.6 p391 #7-8, 13, 22-23 Quiz on Wednesday over 6.3 – 6.5