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 .
