Calculator Input
Formula Used
The calculator uses the transformed trigonometric form:
y = A f(Bx + C) + D
Phase shift = -C / B
Period = base period / |B|
The base period is 2π for sine, cosine, secant, and cosecant in radians. It is π for tangent and cotangent. In degree mode, the matching base periods are 360° and 180°.
Amplitude is |A| for sine and cosine. For tangent, cotangent, secant, and cosecant, |A| is treated as a vertical stretch factor.
How to Use This Calculator
- Select the function family.
- Choose radians or degrees.
- Enter A, B, C, and D from y = A f(Bx + C) + D.
- Enter an x value for a quick function evaluation.
- Set the table start, end, and step values.
- Press Calculate to show the result above the form.
- Use the CSV or PDF button to save the current calculation.
Example Data Table
| Function | A | B | C | D | Phase shift | Period |
|---|---|---|---|---|---|---|
| 2sin(3x - 1.5) + 4 | 2 | 3 | -1.5 | 4 | 0.5 right | 2π / 3 |
| cos(2x + 6) | 1 | 2 | 6 | 0 | 3 left | π |
| 4tan(5x - 10) | 4 | 5 | -10 | 0 | 2 right | π / 5 |
Understanding Phase Shift
Phase shift describes horizontal movement in a transformed periodic function. It tells where the graph starts after the inside expression changes. For sine and cosine, this movement affects peaks, troughs, and intercepts. For tangent and cotangent, it moves asymptotes and center points. The calculator reads the standard form y = A f(Bx + C) + D. It then rewrites the inside as B(x - h). The value h is the phase shift. A positive h moves the graph right. A negative h moves the graph left.
Why It Matters
Phase shift is useful in algebra, precalculus, engineering, sound analysis, and wave modeling. Many signals repeat, but their starting point changes. A shifted sine wave can describe delayed motion. A shifted cosine wave can describe a machine cycle beginning after time zero. The same idea helps compare two waves. It also helps locate graph features without drawing every point.
Calculator Features
This tool accepts six common trigonometric families. You can work in radians or degrees. It reports amplitude, period, phase shift, vertical shift, direction, and a rewritten equation. It can also evaluate the transformed function at a selected x value. A generated table gives ordered pairs over your chosen interval. These rows help confirm a graph or prepare class notes. Export buttons save the current result as a spreadsheet file or a simple document file.
Better Input Habits
Use the coefficient beside x as B. Use the constant inside the function as C. Keep the sign of C exactly as it appears. For example, sin(2x - 4) has C = -4. If the form is sin(2(x - 2)), expand first or enter B = 2 and C = -4. Always choose the correct angle unit. Mixing degree constants with radian periods gives wrong movement.
Reading the Answer
The period explains one full cycle. The phase shift explains horizontal displacement. The vertical shift raises or lowers the midline. The amplitude controls height for sine and cosine. For tangent, cotangent, secant, and cosecant, amplitude is reported as a stretch factor because those functions do not have bounded peaks.
Check the table before exporting. Rounded values can hide exact intercepts, so use formula lines when precision matters in formal study reports.
FAQs
What is phase shift?
Phase shift is the horizontal movement of a transformed periodic graph. In y = A f(Bx + C) + D, it equals -C / B.
Why does the sign change in the answer?
The sign changes because Bx + C is rewritten as B(x - h). Solving that inside expression gives h = -C / B.
Can I use degrees?
Yes. Choose degree mode before calculating. Then C, x values, phase shift, and period are interpreted with degree-based cycles.
What happens when B is zero?
When B is zero, the function has no normal horizontal cycle. The calculator reports that period and phase shift are undefined.
Does amplitude apply to tangent?
Tangent has no maximum or minimum height. The A value works as a vertical stretch factor rather than a bounded amplitude.
How is period calculated?
The base period is divided by |B|. Sine and cosine use 2π or 360°. Tangent and cotangent use π or 180°.
Can this calculator make a table?
Yes. Enter start, end, and step values. The calculator creates x values, inside arguments, base results, and final y values.
What do the export buttons save?
The CSV button saves spreadsheet-friendly rows. The PDF button saves a compact result summary with selected generated table values.