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
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 they 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.
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
10.68
2
11.30
3
11.98
4
5
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