Quick answer

Rewrite in standard form, identify C and B, then phase shift = C/B. Report amplitude, period, and vertical shift separately.

Formula

  • Phase shift = C / B
  • Period = 2π / B
  • Amplitude = A
  • Vertical shift = D

Introduction

Calculation questions rarely ask for phase shift alone. Instructors expect a full transformation list: amplitude, period, midline, and horizontal shift together.

The hardest step is not division but preparation: many missed answers come from leaving the argument factored or from copying a negative sign incorrectly.

Review the phase shift formula article when the problem uses y = A sin(B(x - C)) + D so you know whether the worksheet wants C or C/B.

After hand work, compare every row with the Phase Shift Calculator on our homepage.

What the calculation represents

You are finding the input x where the argument Bx - C equals zero at the reference point of the cycle, relative to the parent sine or cosine graph.

Degree and radian versions differ only in unit labels if you convert consistently before dividing.

Calculator method: type decimals for π when needed, choose sine or cosine, and read the phase shift row without retyping the formula.

For a library of finished problems, phase shift examples shows sine, cosine, and applied waveform cases you can mimic on your assignments.

Calculation template

  • f(x) = A sin(Bx - C) + D
  • Phase shift = C / B
  • Period = 2π / B

Extract horizontal shift only after Bx - C is visible. If the problem gives B(x - h), multiply to get C = Bh.

Identify transformation values in a table: A, B, C, D, then compute derived quantities.

Show units on the final line to avoid losing credit on otherwise correct arithmetic.

Step-by-step method

  1. Standard form. Write A sin(Bx - C) + D or cosine equivalent.
  2. Factor check. Expand B(x - h) if the problem uses parentheses.
  3. List parameters. Record A, B, C, D in one column.
  4. Divide C by B. Simplify the fraction before rounding.
  5. Compute period. Use 2π/B and simplify to a coefficient times π when possible.
  6. Calculator verification. Enter the same numbers into the Phase Shift Calculator.

Radian calculation example

For f(x) = -2 cos(4x - π) + 5, A = -2, B = 4, C = π, D = 5.

Phase shift = π/4 radians. Period = 2π/4 = π/2. The negative amplitude reflects the graph across the midline.

If C had been given as 90° instead of π/2 radians, convert before dividing unless the course allows degree reporting throughout.