Transformations of Cosine Functions - Algebra-2

1. Fundamental Concepts

Definition: Transformations of cosine functions involve changes to the amplitude, period, phase shift, and vertical shift of the basic cosine function $$\cos(x)$$.
Amplitude: The maximum distance from the midline of the function to its maximum or minimum value.
Period: The length of one complete cycle of the cosine function.
Phase Shift: Horizontal shift of the cosine function, indicating how much the graph is shifted left or right from the standard position.
Vertical Shift: Vertical displacement of the entire cosine function up or down.

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

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 $$y = 3 \cdot \cos(2(x - \frac{\pi}{4})) + 1$$.

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 function should yield a value consistent with the transformations applied.

Problem: Graph the function $$y = -2 \cdot \cos(3(x + \frac{\pi}{6})) - 1$$.

Amplitude: $$|-2| = 2$$
Period: $$\frac{2\pi}{|3|} = \frac{2\pi}{3}$$
Phase Shift: $$\frac{-\frac{\pi}{6}}{3} = -\frac{\pi}{18}$$ (to the left)
Vertical Shift: $$-1$$
Graph the function by starting with the basic cosine wave, then apply the transformations in order: amplitude, period, phase shift, and vertical shift.

Validation: Check key points on the graph such as the maximum, minimum, and midline values to ensure they align with the calculated transformations.

Identify Components: Always start by identifying the components $$A$$, $$B$$, $$C$$, and $$D$$ in the general form $$y = A \cdot \cos(B(x - C)) + D$$.
Apply Transformations Sequentially: Apply each transformation step-by-step: amplitude first, then period, followed by phase shift, and finally vertical shift.
Graphing Strategy: Use a table of values to plot key points and sketch the transformed graph accurately.
Check Consistency: Substitute specific values into the function to verify the correctness of the transformations.