正弦和余弦图像变换

Problem

Create a video about trigonometric transformations.

Step 1: Identify the General Form

A transformed sine function can be written as

$$y = A\sin(B(x - C)) + D.$$

Each parameter changes the graph in a specific way.

Step 2: Adjust the Amplitude and Period

The value of $A$ controls the amplitude of the graph. The amplitude is

$$|A|.$$

The value of $B$ controls the period. For a sine function, the period is

$$\frac{2\pi}{B}.$$

Step 3: Apply the Shifts

The value of $C$ creates a horizontal phase shift. The graph shifts horizontally by $C$.

The value of $D$ creates a vertical shift. The graph shifts up or down by $D$.

Step 4: Final Result

By adjusting the amplitude with $A$, the period with $\frac{2\pi}{B}$, the horizontal phase shift with $C$, and the vertical shift with $D$, you can accurately graph any transformed trigonometric function of the form

$$y = A\sin(B(x - C)) + D.$$