Vertical Translation and Phase
Shift
Trigonometry
MATH 103
S. Rook
Overview
• Section 4.3 in the textbook:
– Vertical translation
– Phase shift
– Graphing sine and cosine in general
2
Vertical Translation
How Vertical Translation Affects a
Graph
• The graphs of y = k + A sin x and y = k + A cos x are
related to the graphs y = A sin x and y = A cos x:
– The value of k is added to each y-coordinate of
y = A sin x or y = A cos x to get the new functions
y = k + A sin x or y = k + A cos x
• E.g. (0, 0) on y = sin x would become (0, -2) on the
graph of y = -2 + sin x
• Amplitude = |A|
– The maximum value is k + |A| and the minimum value
is k + -|A|
– The range of y = k + A sin x or y = k + A cos x is then
4
[k + -|A|, k + |A|]
How Vertical Translation Affects a
Graph (Continued)
• If k > 0
y = k + A sin x or y = k + A cos x will be shifted UP
k units as compared to y = A sin x or y = A cos x
• If k < 0
y = k + A sin x or y = k + A cos x will be shifted
DOWN k units as compared to y = A sin x or
y = A cos x
• The value of k affects ONLY the y-coordinate
• The value of k affects ONLY the vertical
translation
5
Graphing y = k + A sin x or
y = k + A cos x
• To graph one cycle of y = k + A sin x or
y = k + A cos x:
– Follow the same steps for graphing y = A sin x or
y = A cos x:
• Dividing intervals, creating a table of values, etc.
– Must take the effect of k into account when
constructing the graph:
• The maximum value will be k + |A| and the minimum
value will be k + -|A|
6
Vertical Translation (Example)
Ex 1: Graph one complete cycle:
y = -3 + 2sin x
7
Phase Shift
Phase Shift
• Phase shift occurs when we move a graph
horizontally
– In the x-direction
• We have a phase shift when we add/subtract
a quantity to the variable inside of the
trigonometric function
– Recall that the inside of a trigonometric function is
also called the argument
• e.g.
 

y  cos  x  
6

9
How Phase Shift Affects a Graph
• Consider the effects of phase shift on y = sin x and
y = sin(x – h) or y = cos x and y = cos(x – h)
• Graph one cycle of y = sin x and
y = sin(x – π⁄2) using a table of values:
– Notice that y = sin x completes one cycle in the interval
0 to 2π and y = sin(x – π⁄2) completes one cycle in the
interval π⁄2 to 5π⁄2
• y = sin(x – π⁄2) has
been shifted to
the right by h = π⁄2
10
How Phase Shift Affects a Graph
(Continued)
– The value of the phase shift coincides with the
leftmost value of a cycle
• The phase shift for y = sin x is 0 in the interval 0 to 2π
and the phase shift for y = sin(x – π⁄2) is π⁄2 in the
interval π⁄2 to 5π⁄2
• To establish a relationship between y = sin x and
y = sin(x – h) or y = cos x and y = cos(x – h):
– When h = 0, the graph begins a cycle at 0
0  x  2
– When h = π⁄2, the graph begins a cycle at π⁄2


5 (Add π⁄2; 2  
 2 
0 x
2
 x
2
2
2

4
2


2

5
)
2
11
Relationship Between h and Phase
Shift
• Therefore, for y = sin(x – h) or y = cos(x – h):
phase shift = h
– If y = sin(x – h) or y = cos(x – h)
h>0
– If y = sin(x + h) or y = cos(x + h)
h<0
– However, if the signs are confusing, we can also find the
value of the phase shift using the interval method:
• i.e. 0 ≤ argument ≤ 2π and then isolate the variable
(e.g. x) to obtain: phase shift ≤ x ≤ end value
12
Graphing y = sin(x – h) or
y = cos(x – h)
• To graph one cycle, we repeat the same steps for
graphing y = A sin x or y = A cos x EXCEPT:
–
–
–
–
Calculate the phase shift
Calculate the subinterval length (π⁄2 if the period is 2π)
Adjust the x-axis to start at the phase shift
Label the x-axis by adding increments of the subinterval
until the end value of the interval is reached
• Note that we have the SAME PERIOD on
the phase shifted interval as we do when using
Period 
2
– Can verify by subtracting the endpoints
– This period is 2π if B = 1 (a constant of 1 multiplying the
13
variable)
B
Phase Shift (Example)
Ex 2: Graph one complete cycle:
a)
 

y  cos  x  
4

b)
 

y  sin  x  
3

14
Graphing Sine and Cosine in
General
Graphing y = sin(Bx + C) or
y = cos(Bx + C)
• The process is slightly different for calculating
phase shift for y = sin(Bx + C) or y = cos(Bx + C)
– Use the interval method:
0  Bx  C  2    C  Bx  2   C  
C
B
 x
2  C
B
• Recall that the phase shift coincides with the
leftmost value of the interval
– Phase shift is then -C⁄B and period is 2π⁄B (subtract
the endpoints of the interval)
– Can either derive the period and phase shift using
the interval method or memorize the formulas:
period = 2π⁄B and phase shift = -C⁄B
16
Summary of y = k + A sin(Bx + C) or
y = k + A cos(Bx + C)
• Given y = k + A sin(Bx + C) or
y = k + A cos(Bx + C), recall that the:
– Vertical translation is k
– Amplitude is |A|
– Period is 2π⁄B
– Phase shift is -C⁄B
• Note that it may be easier to use the interval method to
obtain the period and phase shift
– Domain is (-oo, +oo)
– Range is [k + -|A|, k + |A|]
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Graphing y = k + A sin(Bx + C) or
y = k + A cos(Bx + C)
• To graph y = k + A sin(Bx + C) or y = k + A cos(Bx + C):
– Find the values for A (amplitude), period, k (vertical
translation), and phase shift
– “Construct the Frame” for one cycle:
• Calculate the subinterval length (easiest to use
period⁄ )
4
• x-axis by the interval method:
– Start the cycle at the leftmost value of the interval
(phase shift)
– Label the x-axis by adding increments of the
subinterval until the end value of the interval is
reached
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Graphing y = k + A sin(Bx + C) or
y = k + A cos(Bx + C) (Continued)
• x-axis by the formula method:
– Start the cycle at -C⁄B (phase shift)
– Label the x-axis by adding increments of the
subinterval until 2π⁄B – C⁄B is reached
• y-axis:
– Minimum value is k + -|A|
– Maximum value is k + |A|
• Create a table of values for the points marked on the
x-axis
• Connect the points by using the shape of the sine or
cosine function
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– Extend the graph as necessary
Graphing Sine and Cosine in
General (Example)
Ex 3: a) identify the amplitude b) identify the
vertical translation c) identify the period d)
identify the phase shift e) graph one cycle
 
1
y  3  2 sin  x  
2
2
20
Graphing Sine and Cosine in
General (Example)
Ex 4: a) identify the amplitude b) identify the
vertical translation c) identify the period d)
identify the phase shift e) graph on the given
interval
3 
9
9

y   2  cos   x 
,   x 
3
4 
4
4

1
21
Summary
• After studying these slides, you should be able to:
– Identify the vertical translation, amplitude, period,
and phase shift for ANY sine or cosine graph or
equation
– Graph an equation of the form y = k + A sin(Bx + C) or
y = k + A cos(Bx + C)
• Additional Practice
– See the list of suggested problems for 4.3
• Next lesson
– The Other Trigonometric Functions (Section 4.4)
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