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.