https://file.aiquickdraw.com/mathbot/file/1720739343145y7xkk365.png

To determine which equation matches the given graph, let's analyze the key features of the graph and the given equations.

The graph shows a polynomial function with roots at \(x = -6\), \(x = 0\), and \(x = 2\). We can use these roots to construct the polynomial.

Given roots:
- \(x = -6\)
- \(x = 0\)
- \(x = 2\)

The polynomial can be written as:
\[ f(x) = k(x + 6)x(x - 2) \]

Since there is no vertical stretch or shrink indicated, we can assume \( k = 1 \). Thus:
\[ f(x) = (x + 6)x(x - 2) \]

Now, let's match this with the given options:

- \( f(x) = (x - 6)(x^2 - 2) \)
- \( f(x) = (x + 6)(x^2 + 2) \)
- \( f(x) = (x + 6)(x^2 - 2) \)
- \( f(x) = (x - 6)(x^2 + 2) \)

The correct match is:
\[ f(x) = (x + 6)(x^2 - 2) \]

This corresponds to the third option:
\[ f(x) = (x + 6)(x^2 - 2) \]

Would you like more details or have any questions?

Here are 5 numbered questions you might want to ask:
1. How do you factor polynomials with complex roots?
2. What is the significance of the leading coefficient in a polynomial?
3. How do you determine the degree of a polynomial?
4. Can you explain how polynomial functions behave at infinity?
5. What are the different methods for finding polynomial roots?

**Tip:** Always consider the roots of the polynomial and their multiplicity when analyzing a graph to identify the corresponding equation.

Learn how to match a polynomial equation to a given graph with specified roots. This problem explores polynomial functions and their roots, demonstrating how to construct the polynomial equation based on given graph features. Suitable for Grades 10-12 students and those preparing for advanced algebra.

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