Quick answer

y = A sin(B(x - C)) + D shifts horizontally by C. The expanded form f(x) = A sin(Bx - C) + D shifts by C/B.

Formula

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

Introduction

The core equation y = A sin(B(x - C)) + D packs four transformations into one expression. A controls height, B controls compression of the input, C controls horizontal placement, and D moves the midline.

Courses disagree slightly on whether to report the shift as C or C/B because authors factor the argument differently. Your grade depends on matching the form on the worksheet, not memorizing one sentence from a different book.

If the definition felt abstract, revisit what a phase shift is first, then return here for the algebra that turns words into fractions.

Open the Phase Shift Calculator on our homepage using the expanded form so the phase shift row matches C/B from your factored equation.

Formula explanation and graph interpretation

General sinusoidal form separates shape from position. Amplitude |A| sets peak height above and below the midline y = D unless a course treats signed A as a reflection.

The input scaling B affects period through 2π/B. A larger |B| squeezes more cycles into the same x-interval, which is easy to confuse with phase unless you track each parameter separately.

Graph interpretation workflow: draw the midline, mark amplitude rails, measure one period, then place the first reference point at the phase shift before you connect the curve.

Once parameters are identified, graphing phase shift functions shows how to turn the formula into a sketch you can compare with calculator output.

Sine and cosine transformation lines

  • y = A sin(B(x - C)) + D
  • y = A cos(B(x - C)) + D
  • Expanded: A sin(Bx - BC) + D
  • Phase shift (Bx - C form) = C / B

Cosine transformations follow the same structure. A common rewrite cos(x) = sin(x + π/2) explains why shifted sine and cosine graphs look like slides of one another.

Horizontal shift formula summary: factored form uses C inside the parentheses; expanded form divides by B.

Always state units. Radian answers are standard in calculus-based classes; degree reporting is common in introductory trig.

Apply the formula reliably

  1. Copy the equation exactly. Do not change signs while rewriting.
  2. Choose a target form. Pick factored or expanded layout based on the question.
  3. Identify C correctly. After expansion, C is the constant subtracted inside Bx - C.
  4. Divide by B. Phase shift = C/B for the expanded calculator form.
  5. Compute period and shifts. Report 2π/B, A, and D as separate answers.
  6. Check with technology. Enter the expanded coefficients into the Phase Shift Calculator.

Factored vs expanded example

Given y = 4 sin(2(x - 1)) + 3, the factored shift is 1 unit before expansion.

Expand: 4 sin(2x - 2) + 3, so B = 2 and C = 2, giving phase shift C/B = 1, period π, amplitude 4, vertical shift 3.

Both descriptions refer to the same graph; only the reported shift number changes with form.