Find Zeros of a Transformed Cubic Function

Algebra-2

1. Fundamental Concepts

Definition: A transformed cubic function is a polynomial of degree three that has been modified through operations such as shifting, stretching, or reflecting.
Zeros: The zeros of a function are the values of \(x\) for which the function equals zero.
Transformations: Transformations can affect the location and number of zeros of a cubic function.

2. Key Concepts

Basic Rule: $$f(x) = a(x - h)^3 + k$$

Degree Preservation: The highest degree in the result matches input

Application: Used to model real-world phenomena such as motion and engineering problems

3. Examples

Example 1 (Basic)

Problem: Find the zeros of the function \(f(x) = (x - 2)^3\).

Step-by-Step Solution:

Set the function equal to zero: \((x - 2)^3 = 0\)
Solve for \(x\): \(x - 2 = 0 \Rightarrow x = 2\)

Validation: Substitute \(x = 2\) → Original: \((2 - 2)^3 = 0\); Simplified: \(0 = 0\) ✓

Example 2 (Intermediate)

Problem: Find the zeros of the function \(g(x) = 2(x + 1)^3 - 8\).

Step-by-Step Solution:

Set the function equal to zero: \(2(x + 1)^3 - 8 = 0\)
Solve for \(x\): \(2(x + 1)^3 = 8 \Rightarrow (x + 1)^3 = 4 \Rightarrow x + 1 = \sqrt[3]{4} \Rightarrow x = \sqrt[3]{4} - 1\)

Validation: Substitute \(x = \sqrt[3]{4} - 1\) → Original: \(2(\sqrt[3]{4})^3 - 8 = 2 \cdot 4 - 8 = 8 - 8 = 0\); Simplified: \(0 = 0\) ✓

4. Problem-Solving Techniques

Visual Strategy: Graph the function to visualize the transformations and their effects on the zeros.
Error-Proofing: Double-check each step of the solution process, especially when dealing with cube roots and other complex expressions.
Concept Reinforcement: Practice with various types of transformations to understand how they affect the zeros of a cubic function.