https://file.aiquickdraw.com/mathbot/file/1720739298968n26z120u.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 \(x = -7\), \(x = 0\), and \(x = 7\). We can use these roots to construct the polynomial.

Given roots:
- \(x = -7\)
- \(x = 0\)
- \(x = 7\)

The polynomial can be written as:
\[ f(x) = k(x + 7)x(x - 7) \]

Since there is no vertical stretch or shrink indicated, we can assume \( k = 1 \). Thus:
\[ f(x) = (x + 7)x(x - 7) \]

This simplifies to:
\[ f(x) = x(x^2 - 49) \]
\[ f(x) = x^3 - 49x \]

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

- \( f(x) = (x - 7)(x^2 - 49) \)
- \( f(x) = (x + 7)(x^2 - 49) \)
- \( f(x) = x(x^2 + 49) \)
- \( f(x) = x(x^2 - 49) \)

The correct match is:
\[ f(x) = x(x^2 - 49) \]

This corresponds to the fourth option:
\[ f(x) = x(x^2 - 49) \]

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?
2. What are the different forms of polynomial equations?
3. How do you determine the roots of a polynomial?
4. Can you explain how to graph a polynomial function?
5. How do you identify key features of a polynomial graph?

**Tip:** When analyzing a graph, always look for key features such as roots, intercepts, and turning points to help identify the corresponding equation.

Learn how to match a polynomial equation to a graph by analyzing its roots and features. This detailed explanation covers polynomial functions, roots, and key algebraic concepts. Suitable for high school students.

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