To identify the function that matches the graph, let's analyze the key characteristics:

1. Symmetry: The graph is symmetric about the y-axis, which suggests it could be an even function, meaning terms should have even powers or a structure that allows symmetry in multiplication.

2. Zeros: The graph touches or crosses the x-axis at three distinct points, one of which appears to be the origin. This suggests that $x = 0$, $x = a$, and $x = b$ are roots of the polynomial.

3. Multiplicity: The way the curve behaves at each root indicates the multiplicities:

   • The graph touches the x-axis at $x = 0$ and bounces back, suggesting an even multiplicity.
   • Similarly, at $x = a$ and $x = b$, the curve bounces, also suggesting even multiplicities for these roots.

Considering these features, let’s examine the answer choices:

1. $f(x) = x(x - a)^3(x - b)^3$: This function would give odd multiplicities at $x = a$ and $x = b$, which does not match the graph since we observe even behavior at each root.

2. $f(x) = (x - a)^5(x - b)$: This function only has two roots with odd multiplicities, so it doesn’t match the graph's symmetry or the required root behavior.

3. $f(x) = (x - a)^2(x - b)^4$: This function has even multiplicities for $x = a$ and $x = b$ but lacks a factor for $x = 0$, which is also a root.

4. $f(x) = x(x - a)^6(x - b)^2$: This function includes $x = 0$ as a root (multiplicity 1), $x = a$ with an even multiplicity (6), and $x = b$ with an even multiplicity (2). This aligns with the graph’s symmetry and the behavior at each root.

Conclusion:

The correct answer is:

• $f(x) = x(x - a)^6(x - b)^2$

Would you like a more detailed breakdown of any specific part of this analysis?

Here are some related questions to expand understanding:

1. Why does an even multiplicity at a root cause the graph to "bounce" off the x-axis instead of crossing it?
2. How would the graph change if one of the roots had an odd multiplicity instead?
3. What is the significance of symmetry in determining the nature of the function?
4. Can this analysis method be applied to higher-degree polynomials with more roots?
5. What role does the leading coefficient play in the shape and orientation of polynomial graphs?

Tip: Always examine the behavior at each root to determine multiplicity and symmetry, as these are key to identifying polynomial functions from graphs.