Definition: A phase shift is the horizontal translation of a trigonometric graph; for y = A sin(B(x – h)) the graph moves h units to the right.
What is a phase shift?
When a sine or cosine wave slides left or right, that slide is its phase shift. Write the function as y = A sin(B(x – h)) + k. Then h is the shift: positive moves right, negative moves left. The value k lifts the midline.
How do you find it?
Factor B out of the inside first. If the function is y = 2 sin(3x – pi), rewrite it as 2 sin(3(x – pi/3)). The phase shift is pi/3 to the right, not pi.
Worked example: with period 2 pi/3, the first cycle of this wave starts at x = pi/3 and ends at x = pi/3 + 2 pi/3 = pi.
What is the typical slip?
Reading the shift straight from the constant term. In 3x – pi, the number pi is not the shift until you divide by the coefficient 3. Mathbot suggests setting the inside equal to zero: 3x – pi = 0 gives x = pi/3.
Where to learn it
- Graphing Waves: Sine and Cosine Models – lesson
- Graphing Sine, Cosine and Tangent Functions – lesson