Transformations of Tangent Functions - Algebra-2

1. Fundamental Concepts

Definition: Transformations of tangent functions involve changes in the amplitude, period, phase shift, and vertical shift of the basic tangent function $$y = \tan(x)$$.
Amplitude: The tangent function does not have a traditional amplitude because it extends infinitely in both directions. However, transformations can affect its behavior.
Period: The period of the tangent function is $$\pi$$. Transformations can alter this period.
Phase Shift: This is a horizontal shift of the graph, which can be determined by solving for the value that shifts the function left or right.
Vertical Shift: This is a vertical translation of the graph, moving it up or down.

2. Key Concepts

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

Amplitude: The tangent function does not have an amplitude in the traditional sense, but transformations can affect its behavior.

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

Phase Shift: The phase shift is given by $$C$$ (to the right if positive, to the left if negative).

Vertical Shift: The vertical shift is given by $$D$$ (upward if positive, downward if negative).

3. Examples

Example 1 (Basic)

Problem: Graph the function $$y = 2 \cdot \tan(2x - \pi) + 1$$.

Step-by-Step Solution:

Identify the parameters: $$A = 2$$, $$B = 2$$, $$C = \frac{\pi}{2}$$, $$D = 1$$.
The period is $$\frac{\pi}{|2|} = \frac{\pi}{2}$$.
The phase shift is $$\frac{\pi}{2}$$ to the right.
The vertical shift is 1 unit up.

Validation: Check key points on the graph to ensure they align with the transformations applied.

Example 2 (Intermediate)

Problem: Determine the equation of the tangent function with a period of $$\pi$$, a phase shift of $$-\frac{\pi}{4}$$, and a vertical shift of 3 units down.

Step-by-Step Solution:

Given the period is $$\pi$$, $$B = 1$$.
Given the phase shift is $$-\frac{\pi}{4}$$, $$C = -\frac{\pi}{4}$$.
Given the vertical shift is 3 units down, $$D = -3$$.
The equation is $$y = \tan(x + \frac{\pi}{4}) - 3$$.

Validation: Substitute specific values of $$x$$ into the equation to verify the transformations.

4. Problem-Solving Techniques

Graphical Analysis: Use graphs to visualize the transformations and their effects on the tangent function.
Substitution Method: Substitute known values into the transformed function to check the correctness of the transformations.
Step-by-Step Approach: Break down the problem into identifying each transformation parameter and applying them sequentially.