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 = 3sinθ + 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 = 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?

1. Find the amplitude, “A”: (max – min)/2

2. Find the vertical translation, “h”: (max + min)/2

3. Find “k”: Solve 2π/k = Period

4. 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