Last updated: August 5, 2026
Phase Shift Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
In the standard transformed trig form y = a·f(bx + c) + d, the phase shift is −c/b. This calculator also reports amplitude, period, and vertical shift for sine, cosine, and tangent models in either degrees or radians. It helps translate a symbolic trig expression into an interpretable graph transformation.
For y equals a times f of bx plus c plus d, the phase shift is negative c divided by b.
Key Takeaways
- Phase shift is a horizontal translation, not a vertical one.
- Always divide the inside constant by the b coefficient to find the shift.
- Sine and cosine use a different base period than tangent.
- Amplitude comes from |a| for sine and cosine transformations.
- Unit consistency matters because degrees and radians express the same shift differently.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
For y = a·f(bx + c) + d, amplitude = |a|, period = basePeriod/|b|, and phase shift = −c/b
Where:
- a=vertical stretch factor
- b=horizontal frequency factor
- c=phase constant(degrees or radians)
- d=vertical shift
Worked Examples
Sine in degrees
For y = 2sin(3x − 90) + 1.
- 1Amplitude = |2| = 2.
- 2Period = 360/3 = 120°.
- 3Phase shift = -(-90)/3 = 30° to the right.
Cosine in radians
For y = -4cos(0.5x + π) − 2.
- 1Amplitude = 4.
- 2Period = 2π / 0.5 = 4π.
- 3Phase shift ≈ -6.283186, meaning 2π left.
Tangent transformation
For y = tan(2x + 30).
- 1Tangent uses base period 180°.
- 2Period = 180/2 = 90°.
- 3Phase shift = -30/2 = -15°, so the graph shifts left 15°.
Introduction
Phase shift describes how far a trigonometric graph moves horizontally from its parent function. In transformed models such as y = a·sin(bx + c) + d or y = a·cos(bx + c) + d, the same coefficients also control amplitude, period, and vertical shift. This calculator untangles those roles, works in degrees or radians, and helps students interpret the transformation instead of memorizing disconnected rules.
What phase shift means
A phase shift is a horizontal translation. If the graph moves to the right, the phase shift is positive in the usual interpretation used here; if it moves to the left, the shift is negative. Because the constant sits inside the function with the x-term, it must be divided by the horizontal scaling coefficient rather than read directly. That is why the relationship is phase shift = −c/b rather than simply “look at c.”
How a, b, c, and d work together
The coefficient a controls vertical stretch or reflection, b controls horizontal compression and period, c controls horizontal translation, and d moves the midline vertically. Those effects happen simultaneously, so a transformed trig equation often looks more complicated than it really is. Breaking the expression into these four jobs helps you read the graph systematically instead of treating each new equation as a brand-new problem.
Period depends on the trig function
Sine and cosine start with a base period of 360° or 2π, while tangent starts with a base period of 180° or π. Dividing that base period by |b| gives the transformed period. The calculator changes the period rule automatically when you switch function types, which keeps the interpretation aligned with the chosen trig family.
Degrees versus radians
The same transformation ideas work in both units, but the numeric values look different. A 30° shift is the same as π/6 radians, and a 120° period is the same as 2π/3 radians. Problems in precalculus may use degrees for graphing clarity, while calculus and higher trigonometry often prefer radians because they connect more directly to circular motion and analytic formulas.
Worked example
For y = 2sin(3x − 90) + 1, the amplitude is 2, the period is 360/3 = 120°, the phase shift is −(−90)/3 = 30° to the right, and the vertical shift is +1. Seeing all four values together helps explain the full graph: it oscillates twice as tall as the parent curve, cycles more quickly, starts later on the x-axis, and rides on a midline one unit above zero.
Common errors
The biggest mistake is forgetting to divide c by b when reading the horizontal shift. Another is forgetting the negative sign in phase shift = −c/b. Students also mix up period and amplitude or assume tangent has the same base period as sine. Writing the function in the standard a·f(bx + c) + d form before reading values is one of the safest habits you can build.
Applications
Phase shifts matter in waves, electrical signals, seasonal models, oscillations, sound synthesis, and any periodic pattern. In applied settings, a horizontal shift can represent a delay, a lead, or a change in starting point rather than a purely visual graph move. That interpretation is why phase shift is central in both mathematics and science, not just in graphing exercises.
Reading the graph after the calculation
Once you know amplitude, period, phase shift, and vertical shift, you can sketch the transformed function far more confidently. The phase shift tells you where a familiar landmark such as a sine crossing or cosine peak begins, the period tells you how long one cycle lasts, and the vertical shift tells you where the midline sits. Together, those values summarize the graph efficiently.
Quick Reference Card
Phase shift cheat sheet
Quick reference • Phase Shift Calculator
Phase shift = −c / b for y = a·f(bx + c) + dValid range: b must be nonzero
Common Values
⚠ Watch Out
- •Do not read c directly without dividing by b.
- •Keep the negative sign in the phase-shift formula.
- •Use tangent's base period π or 180°, not 2π or 360°.
- •Stay consistent with degrees or radians from start to finish.
Pro Tips
- →Rewrite the function in standard form before interpreting it.
- →Use absolute value on a when you want amplitude magnitude.
- →Check the period first to understand the graph's horizontal scale.
- →Describe the shift as left or right after computing the signed value.
FAQs
What is phase shift?
Phase shift is the horizontal displacement of a trig graph relative to its parent function.
Why is the phase shift formula negative c over b?
Because the horizontal translation is hidden inside the grouped expression bx + c, so solving bx + c = 0 isolates the shift as −c/b.
Does tangent use the same period formula as sine?
No. Tangent has base period 180° or π, while sine and cosine use 360° or 2π.
What does amplitude mean?
Amplitude is the vertical distance from the midline to a peak for sine and cosine. It is |a| in the transformed form.
Can tangent have an amplitude?
Not in the same sense as sine and cosine, because tangent does not have a bounded maximum or minimum. The calculator still reports |a| as the vertical scaling factor.
How do I tell left from right shift?
A positive computed phase shift means the graph moves right, while a negative one means the graph moves left.
Should I use degrees or radians?
Use the unit required by your course, graph, or model. The calculator supports both as long as you stay consistent.