Question

State the transformations, and graph the following trigonometric functions from 0 to 2 π

f(x) = -1/4 sin(2x-π)+1

Answer 1

The function involves horizontal compression by a factor of 2, a horizontal shift to the right by π/2, a vertical stretch by -1/4, and a vertical shift upwards by 1 unit. These transformations are applied to the basic sine function to plot the graph from 0 to 2π.

Step-by-step explanation:

The function f(x) = -\frac{1}{4}\sin(2x-\pi)+1 involves several transformations of the basic sine function. The transformations are as follows:

1. Horizontal compression by a factor of \(2\) due to the \(2x\) inside the sine function.
2. Horizontal shift to the right by \(\frac{\pi}{2}\) because of the \(-\pi\) added to \(2x\).
3. Vertical stretch by a factor of \(-\frac{1}{4}\), which also reflects the graph over the x-axis.
4. Vertical shift upwards by \(1\) unit.

To graph this function from \(0\) to \(2\pi\), start with the basic sine graph, apply these transformations step by step, and plot the resulting points to get the transformed wave.