Transformations of Sine Functions - Algebra-2

1. Fundamental Concepts

Definition: Transformations of sine functions involve changes to the amplitude, period, phase shift, and vertical shift of the basic sine function $$y = \sin(x)$$.
Amplitude: The maximum distance from the midline of the sine wave to its peak or trough.
Period: The length of one complete cycle of the sine wave.
Phase Shift: Horizontal shift of the sine wave along the x-axis.
Vertical Shift: Vertical displacement of the sine wave along the y-axis.

General Form: $$y = A \cdot \sin(B(x - C)) + D$$

Amplitude (A): The amplitude is the absolute value of $$A$$.

Period (B): The period is given by $$\frac{2\pi}{|B|}$$.

Phase Shift (C): The phase shift is given by $$\frac{C}{B}$$.

Vertical Shift (D): The vertical shift is $$D$$.

Problem: Identify the amplitude, period, phase shift, and vertical shift for the function $$y = 3 \cdot \sin(2(x - \frac{\pi}{4})) + 1$$.

Amplitude: $$|A| = |3| = 3$$
Period: $$\frac{2\pi}{|B|} = \frac{2\pi}{2} = \pi$$
Phase Shift: $$\frac{C}{B} = \frac{\frac{\pi}{4}}{2} = \frac{\pi}{8}$$ (to the right)
Vertical Shift: $$D = 1$$

Validation: Substituting $$x = \frac{\pi}{4}$$ into the original equation should yield a value that reflects the transformations correctly.

Problem: Graph the function $$y = -2 \cdot \sin(\frac{1}{2}(x + \pi)) - 1$$.

Validation: Plotting key points and ensuring the graph reflects the specified transformations.

Identify Components: Break down the function into its components (amplitude, period, phase shift, vertical shift).
Graphing Strategy: Use the identified components to sketch the graph accurately.
Substitution Method: Substitute specific values of $$x$$ to verify the correctness of transformations.