Understanding Sine Transformations
What Changes in a Sine Graph
A sine transformation changes the parent curve y = sin x. The calculator studies each part of the transformed model. It uses amplitude, period, phase shift, and vertical shift. These values explain the shape before any graph is drawn.
Amplitude and Range
Amplitude is the height from the midline to a peak. A larger absolute A makes the wave taller. A negative A reflects the curve across the midline. The range moves with the vertical shift. So the lowest value is D minus amplitude. The highest value is D plus amplitude.
Period and Horizontal Scale
The B value controls horizontal scale. When B grows, the cycle becomes shorter. When B is between zero and one, the cycle becomes longer. A negative B reverses horizontal direction. The calculator still uses absolute B for period. This keeps the period positive and easy to read.
Phase and Vertical Shift
The C value shifts the curve left or right. In this tool, the model is written as A sin(B(x - C)) + D. A positive C shifts the graph right. A negative C shifts it left. The D value moves the midline up or down.
Table and Target Solving
Students can use this page to compare two views. The first view is a direct value at one x input. The second view is a generated table over one full cycle. The table helps reveal the repeating pattern. It also shows peaks, troughs, and midline crossings.
The optional target value is useful for equation solving. It estimates x values where the transformed curve reaches that target. The answers repeat every period. This helps when checking homework or sketching repeated waves.
Practical Tips
Use degrees for school graphing problems written with 360 degrees. Use radians for calculus, physics, and unit circle work. Keep C and x in the same angle unit. Use enough decimals when values are small. Download the table when you need a saved copy.
For reports, the CSV file keeps rows structured. The PDF summary is better for quick sharing. Both options use the current result, table, and inputs. Recalculate after any edit before downloading the final output. This avoids old saved values.
This calculator is only a numeric aid. Always review signs in the original equation. Small sign changes can move the graph differently. Check the model form before submitting.