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Shifting Sine and Cosine Waves: Amplitude, Vertical Shift, Horizontal Shift (Phase Shift), Study notes of Signals and Systems

How to shift the graphs of sine and cosine waves both vertically and horizontally. The horizontal shift, also known as phase shift, is discussed in detail with examples. Constants added inside the parentheses of the angle affect the phase shift for sine and cosine waves.

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1. Phase Shift
2. You should be familiar with the graphs of y= sin xand y= cos x, and how they can be
stretched to change the amplitude and period.
In this lesson, we will shift the wave graphs, both vertically and horizontally. The horizontal
shift is called phase shift.
3. In this example the function 2 sin x+ 3 is the sine wave with amplitude 2, shifted up 3. We
begin with the standard sine wave.
4. We then make the amplitude 2.
5. And finally, the plus 3 will move the graph up 3. The tops of the waves move from 2 up to 5,
and the bottoms of the waves move from negative 2, up 3 to 1. The centerline of the wave is
at 3. The key points on the final graph have the standard x-values of 0, π
2,π,3π
2and 2π, and
y-values 3,5,3,1,3 respectively.
6. The other addition we need to be concerned with is the constant Cadded inside the paren-
theses to the angle. Constants added inside the parentheses will shift the graph left and right.
For sine and cosine waves, this shift is called the phase shift.
7. In this example, we have a sine wave whose frequency is four times that of a standard sine
wave, and the π
4will shift the graph horizontally. Again, we begin with a standard sine wave.
8. We can then adjust the period. The period is 2π
B, which in this case is 2π
4or π
2.
9. The red graph goes through a full wave every π
2, and makes four full waves in reaching 2π.
The π
4will shift the graph horizontally. Again, this shift is called the phase shift. One way to
find the phase shift is to find the value for xthat makes the angle 0. In this case, the angle
is 4x+π
4.
10. We set the angle equal to zero and solve for x. In this case, the phase shift is
π
16 . This will
cause us to shift the graph π
16 to the left.
11. Returning to the unshifted graph. The period is π
2.
12. To mark the standard points, we next find the quarter marks by dividing the period by 4.
13. The quarter marks happen every π
8. The sine wave is at the top at π
8, back to the middle at
2π
8or π
4, at the bottom at 3π
8, and back to the middle at 4π
8, which is π
2.
14. Recall that we found the phase shift to be
π
16 .
15. To help mark the x-axis, it is convenient to find a common denominator between the quarter
marks and the phase shift. In this case, the common denominator is 16. We can then mark
the x-axis every π
16 . Lets restate the key points on the sine wave using the denominator 16.
The wave hits the high point at 2π
16 , back to the middle at 4π
16 , down to the bottom at 6π
16 and
back to the middle at 8π
16 .
pf3

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  1. Phase Shift
  2. You should be familiar with the graphs of y = sin x and y = cos x, and how they can be stretched to change the amplitude and period. In this lesson, we will shift the wave graphs, both vertically and horizontally. The horizontal shift is called phase shift.
  3. In this example the function 2 sin x + 3 is the sine wave with amplitude 2, shifted up 3. We begin with the standard sine wave.
  4. We then make the amplitude 2.
  5. And finally, the plus 3 will move the graph up 3. The tops of the waves move from 2 up to 5, and the bottoms of the waves move from negative 2, up 3 to 1. The centerline of the wave is at 3. The key points on the final graph have the standard x-values of 0, π 2 , π, 32 π and 2π, and y-values 3,5,3,1,3 respectively.
  6. The other addition we need to be concerned with is the constant C added inside the paren- theses to the angle. Constants added inside the parentheses will shift the graph left and right. For sine and cosine waves, this shift is called the phase shift.
  7. In this example, we have a sine wave whose frequency is four times that of a standard sine wave, and the π 4 will shift the graph horizontally. Again, we begin with a standard sine wave.
  8. We can then adjust the period. The period is (^2) Bπ , which in this case is 24 π or π 2.
  9. The red graph goes through a full wave every π 2 , and makes four full waves in reaching 2π. The π 4 will shift the graph horizontally. Again, this shift is called the phase shift. One way to find the phase shift is to find the value for x that makes the angle 0. In this case, the angle is 4x + π 4.
  10. We set the angle equal to zero and solve for x. In this case, the phase shift is − 16 π. This will cause us to shift the graph 16 π to the left.
  11. Returning to the unshifted graph. The period is π 2.
  12. To mark the standard points, we next find the quarter marks by dividing the period by 4.
  13. The quarter marks happen every 2 π π 8. The sine wave is at the top at π 8 , back to the middle at 8 or^ π 4 , at the bottom at^38 π , and back to the middle at^48 π , which is^ π 2.
  14. Recall that we found the phase shift to be − 16 π.
  15. To help mark the x-axis, it is convenient to find a common denominator between the quarter marks and the phase shift. In this case, the common denominator is 16. We can then mark the x-axis every 16 π. Lets restate the key points on the sine wave using the denominator 16. The wave hits the high point at 216 π , back to the middle at 416 π , down to the bottom at 616 π and back to the middle at 816 π.
  1. We now need to shift the wave 16 π to the left. The start of the wave moves to − 16 π , then to the top at 7 π 16 π , back to the middle at 316 π , down to the bottom at 516 π , and back to the middle at
    1. We can extend the wave further if we wish.
  2. Let’s review the process for drawing sine and cosine waves. First, mark the y-axis as follows: D determines the vertical shift, which is where we place the centerline. From the centerline, go up and down according to the amplitude.
  3. Then we mark the x-axis. First, find the period, (^2) Bπ , divide by four to get the quarter marks, and find the phase shift by setting the angle to zero and solving for x. Next, find a common denominator between the quarter marks and the phase shift and mark the x-axis evenly. You can then mark the reference points for the unshifted graph, and then use the phase shift to move the graph left or right.
  4. Heres a full example. It is a cosine wave, so our instinct is to follow the pattern top, middle, bottom, middle, top. Except this cosine wave has a negative sign in front, which flips the graph top to bottom, so it will go bottom, middle, top, middle, bottom. It has an amplitude of 6.
  5. A vertical shift of 2
  6. So the centerline will be at 2, the top will be 6 above 2 which is 8, and the bottom will be 6 below 2 which is -4.
  7. The period is (^2) Bπ , which in this case is 22 π , which is π.
  8. The quarter marks happen every π 4.
  9. To find the phase shift, we set the angle, 2x − π 3 = 0 and solve for x. The phase shift is π 6
  10. We find a common denominator between π 4 and π 6 , which is 12, and we rewrite the quarter marks and the phase shift in terms of π 12 π. When we mark the x-axis, we will mark it every graph.^12. You may wish to jot down this information to refer to as we move ahead to drawing the
  11. Lets mark the y-axis first. The centerline is at 2, the bottom at negative 4 and the top at 8.
  12. We mark the 6 π x-axis every 12 π , emphasizing the quarter marks at π 4 (which is 312 π ), π 2 (which is 12 ),^34 π (which is^912 π ) and^ π^ (which is^1212 π ).
  13. The reference points for a negative cosine wave are placed at the bottom, middle, top, middle and bottom.
  14. We can then draw the unshifted graph, extending the wave as far as we wish by following the pattern.
  15. Finally, we shift the reference points by the phase shift, which was π 6 , or 212 π. We can then draw the final graph.
  16. Now lets go backward. Given the graph, can we find an equation?