Phase Shift Calculator
Analyze a sine or cosine function in the form f(x) = A × trig(Bx – C) + D, inspect its transformed wave, and export the current calculation to a validated Excel workbook.
Function inputs
f(x) = 0.5 × sin(2x – 3) + 4
Results
1.5 units to the right
Transformed wave across two periods
The curve spans x = – 1.641593 to 4.641593 and oscillates around the midline y = 4 from 3.5 to 4.5.
One-period sample points
| Position | x | Angle B(x – shift) | Base trig value | f(x) |
|---|
The nine rows divide one complete period into eighths. The table and Excel workbook use the same canonical values as the graph and result cards.
How to use the phase shift calculator
What this calculator does
This calculator analyzes a sinusoidal function written as f(x) = A × sin(Bx – C) + D or f(x) = A × cos(Bx – C) + D. It finds the horizontal phase shift, amplitude, period, vertical shift, output range, and an optional value at a chosen x-coordinate. It also draws an original two-period graph and lists nine points from one period. The results are exact identities for the entered algebraic model; they do not infer coefficients from a photographed graph or fit noisy observations.
When to use it
Use the calculator when checking a trigonometry exercise, translating an equation into graph features, comparing sine and cosine models, or preparing a clean set of values for a report or spreadsheet. It is also useful for inspecting how one coefficient changes a periodic signal while the others remain fixed.
How to calculate
- The calculator opens with a complete demonstration: Sine, A = 0.5, B = 2, C = 3, D = 4, and x = 1.5. The results, graph, table, and validated Excel workbook are available immediately.
- Select Trigonometric function, then replace coefficients A, B, C, and D with the values from your equation. Use a decimal point for decimals; grouping commas are accepted only in standard thousands positions. Scientific notation and decimal commas are rejected to prevent ambiguous interpretation.
- Optionally enter Evaluate at x. Results update live. Read the phase-shift direction, amplitude, period, midline, range, point value, graph, and one-period table together.
- Select Download Excel to build a fresh workbook from the current validated model. Reset clears the demonstration and all calculated content; Excel export remains disabled until a complete valid set of required coefficients is entered again.
Input guide
Trigonometric function is required and accepts Sine or Cosine. It changes the base wave but not the formulas for phase shift, amplitude, period, or vertical shift. Coefficient A is a required nonzero real number, such as 0.5; its absolute value is the amplitude, while a negative sign reflects the graph across its midline. Coefficient B is required and nonzero, such as 2; larger |B| shortens the period, and a negative value reverses horizontal orientation without making the period negative. Coefficient C is a required real number, such as 3; together with B it sets the phase shift C/B. Coefficient D is a required real number, such as 4; it raises or lowers the midline and the entire range. Evaluate at x is optional, such as 1.5; leaving it blank removes only the point-value result, not the main analysis. A common mistake is reading the shift as C alone or forgetting that the displayed equation uses Bx – C.
Output guide
Phase shift is C/B in horizontal x-units; positive means right, negative means left, and zero means no horizontal translation. Amplitude is |A| and is always nonnegative. Period is 2π/|B| and is always positive. Vertical shift is D, which is also the graph's midline. Range is [D – |A|, D + |A|]. Function value evaluates the selected sine or cosine formula at the optional x-coordinate. Equation restates the current model. The graph's blue curve is f(x), the teal dashed line is the midline y = D, and the table columns report the one-period position, x-coordinate, internal angle, base sine/cosine value, and transformed f(x).
Worked example
For the startup function f(x) = 0.5 × sin(2x – 3) + 4, the phase shift is C/B = 3/2 = 1.5 units right. The amplitude is |0.5| = 0.5, the period is 2π/|2| = π ≈ 3.141593, and the vertical shift is 4, so the range is [3.5, 4.5]. At x = 1.5, the internal angle is 2(1.5) – 3 = 0; therefore f(1.5) = 0.5 × sin(0) + 4 = 4. These values match the first-open result cards, graph, table, and workbook checkpoints.
Learn more
For a textbook treatment of the same coefficient relationships, review LibreTexts' guide to graphs of sine and cosine functions, which states the standard amplitude, period, phase-shift, and midline rules.
How the transformation works
The outside coefficients control vertical behavior: A scales the distance from the midline, and D moves the midline itself. The inside expression controls horizontal behavior. Factoring B gives Bx – C = B(x – C/B), so the translated reference point is x = C/B. Meanwhile, a full sine or cosine cycle requires an internal angle change of 2π, which takes an x-change of 2π/|B|. LibreTexts' focused explanation of phase shift as horizontal displacement provides additional worked transformations.
Negative A does not create a negative amplitude; it flips peaks and troughs around the midline. Negative B does not create a negative period; the absolute value preserves the positive cycle length. The sign of C/B still determines the shift direction. A compact formal definition of amplitude, period, and phase shift confirms these absolute-value conventions.