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Phase Shift

A free Trigonometry lesson from the “Graphs of Trig Functions” unit, with a worked example and practice problems including step-by-step solutions.

The phase shift is the horizontal slide of a sine or cosine graph, read from the form y = a*sin(b(x - h)) + d as the value h. The graph shifts right when the inside is x - h and left when it is x + h. When b is not 1 you must factor b out of the inside first, so the shift is found from b(x - h), not from the raw constant.

What you'll learn

Why it matters: Tide tables and daily temperature models use a phase shift to line up the peak of the wave with the actual clock time of high tide or the hottest hour, instead of forcing it to occur at time zero.

Worked example

Problem. Find the phase shift (amount and direction) of y = 4*sin(2x - pi).

  1. Factor b = 2 out of the inside: 2x - pi = 2(x - pi/2).
  2. Now it matches b(x - h) with h = pi/2.
  3. Because the inside is x - h, the shift is pi/2 to the right.

Answer: right pi/2

Practice problems

1. Find the phase shift (amount and direction) of y = sin(x - pi/3).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the phase shift (amount and direction) of y = sin(x - pi/3).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The inside is x - h with h = pi/3.
  4. Since it is x - h, the graph shifts right.
  5. Phase shift: pi/3 to the right.
  6. Check the result by substituting or estimating: the response should match right pi/3 and make sense in the original problem.

Answer: right pi/3

2. Find the phase shift (amount and direction) of y = cos(x + pi/4).

Show solution
  1. The inside x + pi/4 equals x - (-pi/4), so h = -pi/4.
  2. A plus sign means the graph shifts left.
  3. Phase shift: pi/4 to the left.

Answer: left pi/4

3. In y = a*sin(b(x - h)) + d, a positive value of h shifts the graph...

Choices: to the right · to the left · up · down

Show solution
  1. The form x - h with positive h matches a right shift.
  2. Subtracting inside moves the graph in the positive x direction.
  3. So a positive h shifts right.

Answer: to the right

4. Find the phase shift (amount and direction) of y = 3*sin(x - pi/2) + 1.

Show solution
  1. Look only at the inside: x - pi/2, so h = pi/2.
  2. The minus sign gives a right shift.
  3. Phase shift: pi/2 to the right (a and d do not affect it).

Answer: right pi/2

5. Find the phase shift (amount and direction) of y = cos(2x - pi).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the phase shift (amount and direction) of y = cos(2x - pi).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Factor b = 2: 2x - pi = 2(x - pi/2).
  4. Now h = pi/2 and the inside is x - h.
  5. Phase shift: pi/2 to the right.
  6. Check the result by substituting or estimating: the response should match right pi/2 and make sense in the original problem.

Answer: right pi/2

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