Calculator Input
Formula Used
General model: y = A · f(B(x − C)) + D
A: vertical stretch, compression, or reflection.
B: horizontal frequency factor. Period equals parent period divided by |B|.
C: phase shift. In this form, positive C moves the graph right.
D: vertical shift and midline value.
Parent periods: sin, cos, sec, and csc use 360° or 2π. Tan and cot use 180° or π.
How to Use This Calculator
- Select the parent trigonometric function.
- Enter A, B, C, and D for the transformed model.
- Choose degrees or radians for the x-axis.
- Set the x interval and sample count.
- Use y clipping to keep tangent and reciprocal graphs readable.
- Press the calculate button to view results above the form.
- Download the CSV table or PDF report when needed.
Example Data Table
This table shows example inputs for common transformed trigonometric functions.
| Function | A | B | C | D | Unit | Expected feature |
|---|---|---|---|---|---|---|
| sin | 2 | 1 | 30 | 1 | Degrees | Amplitude 2, shifted right 30, midline y = 1 |
| cos | -1.5 | 2 | 0 | -2 | Radians | Reflected cosine with shorter period |
| tan | 1 | 0.5 | 45 | 0 | Degrees | Wider tangent branches with vertical asymptotes |
| sec | 3 | 1 | 0 | 2 | Degrees | Reciprocal cosine range outside shifted bounds |
Graphing Trig Functions With Transformations
A trigonometric graph shows repeated motion. It can model waves, rotations, sound, tides, and alternating current. This calculator focuses on transformed graphs. You can change amplitude, period, phase shift, and vertical shift. The graph updates after submission. The numeric table helps confirm the shape.
Start with a parent function. Sine and cosine create smooth waves. Tangent and cotangent create repeating branches. Secant and cosecant create curves with vertical gaps. The coefficient A changes vertical stretch and reflection. The coefficient B changes cycle length. The value C moves the graph left or right. The value D moves the midline.
Good graphing needs a useful domain. A small interval gives detail. A wide interval shows repetition. Use degrees when class work uses 0 to 360. Use radians for calculus, physics, and advanced modeling. Increase sample points for smoother curves. Reduce them for quick checks.
Asymptotes are important for tangent, cotangent, secant, and cosecant. These lines mark places where the function is undefined. The tool lists visible asymptotes inside your selected interval. It also clips extreme values on the chart. This makes the picture readable without hiding the actual calculation logic.
The result panel summarizes the main features. It reports period, midline, phase shift, range, sample minimum, sample maximum, zeros, and intercepts. The Plotly chart gives an interactive view. You can zoom, pan, and inspect points. The CSV export is useful for spreadsheets. The PDF export is useful for class notes and reports.
Always compare the formula with the chart. A negative A reflects the curve. A negative B reverses horizontal direction. A larger absolute B makes cycles shorter. A smaller absolute B makes cycles longer. A positive C shifts the model right in this form. A positive D lifts the entire curve. These simple checks prevent most graphing mistakes.
For deeper study, compare two settings at a time. Keep all other values fixed. Watch how one change alters peaks, troughs, crossings, and gaps. This habit builds strong intuition. It also helps when checking homework, designing lessons, or explaining wave behavior to students. Save the exported table when you need exact points for later graph comparison or a written solution during exam review.
FAQs
1. What does A do in a trig graph?
A controls vertical stretch and reflection. For sine and cosine, |A| is the amplitude. A negative value reflects the graph across its midline.
2. What does B do in the formula?
B changes the period. A larger |B| makes cycles shorter. A smaller |B| makes cycles longer. The period equals the parent period divided by |B|.
3. What is the phase shift?
In y = A f(B(x − C)) + D, C is the phase shift. A positive C shifts the graph right. A negative C shifts it left.
4. Why are tangent values clipped?
Tangent and reciprocal functions grow very large near asymptotes. Clipping keeps the graph readable. The undefined positions are still listed as asymptotes.
5. Should I use degrees or radians?
Use degrees for school problems written with 360° cycles. Use radians for calculus, physics, and formulas involving π. Match the unit used in your problem.
6. Why is amplitude not shown for tangent?
Tangent, cotangent, secant, and cosecant do not have a fixed maximum and minimum like sine or cosine. So their amplitude is not fixed.
7. How are zeros estimated?
The calculator scans sampled points for sign changes. It then estimates where the curve crosses the x-axis. More sample points can improve the estimate.
8. Can I export the results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a report containing the formula, chart, summary, and preview table.