Formula and Method for Phase Shift
A sinusoidal function is usually written in the standard form y = A sin(Bx − C) + D (or the cosine version, y = A cos(Bx − C) + D). Each coefficient controls one visual feature of the graph: A sets the amplitude, B sets how quickly the wave repeats, C together with B sets the phase shift, and D sets the vertical shift. The phase shift is the horizontal distance the parent sine or cosine curve has been slid left or right, and it equals C / B — not C by itself.
How the calculation works
Enter A, B, C, and D from your equation, choose whether C is measured in radians or degrees, and pick sine or cosine (the phase-shift math is identical for both). The calculator finds the amplitude as |A|, the period as 2π / |B| (or 360° / |B| in degrees), and the phase shift as C / B. A positive phase shift means the graph is shifted to the right of the parent curve; a negative value means it is shifted to the left. This falls out of setting the argument equal to zero: Bx − C = 0 when x = C/B, which is where the shifted wave begins its cycle. The vertical shift D is simply read off — it is the y-value of the curve's midline.
Common mistakes
- Forgetting to divide by B: in y = sin(2x − π), the phase shift is π/2, not π — always divide C by B, don't read C alone.
- Sign errors: if the equation is written as y = A sin(Bx + C) + D, rewrite it as Bx − (−C) first; the phase shift is then −C/B, the opposite sign of what a careless reading would suggest.
- Mixing radians and degrees: a period of 2π only applies when x is in radians; if the problem uses degrees, the period is 360°/|B| instead — keep the angle unit consistent throughout.
Real-world applications
- Electrical engineering uses phase shift to describe the lag or lead between AC voltage and current waveforms of the same frequency.
- Sound and signal processing use phase shift to align or compare waveforms, such as syncing audio channels or detecting Doppler-shifted signals.
- Physics and engineering use it to model oscillations, tides, and periodic motion that start partway through a cycle rather than at a peak or zero crossing.
- Trigonometry and precalculus courses use phase shift, alongside amplitude and period, to graph transformed sine and cosine curves from their equations.
Frequently Asked Questions
What is the formula for phase shift?
For a sinusoidal function written as y = A sin(Bx − C) + D (or the cosine equivalent), the phase shift equals C/B. It measures how far the graph is shifted left or right compared to the parent sine or cosine curve.
Does a positive phase shift move the graph left or right?
In the standard form y = A sin(Bx − C) + D, a positive value of C/B shifts the graph to the right, and a negative value shifts it to the left. The argument Bx − C equals zero at x = C/B, which is where the shifted cycle begins.
How do I find the phase shift if my equation is written as y = A sin(Bx + C) + D instead?
Rewrite Bx + C as Bx − (−C) first, so the phase shift is −C/B. Adding C inside the parentheses shifts the graph in the opposite direction from an equation written with a minus sign, so watch the sign carefully.
How is the period related to the phase shift?
Period equals 2π/|B| in radians (or 360°/|B| in degrees) and depends only on B, independent of the phase shift C/B. B stretches or compresses the graph horizontally, while C/B slides it left or right without changing its width.
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
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General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
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| Needs | 50% | Rent, groceries, utilities, insurance |
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Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |