Let's analyze the given graph and the function options to identify the correct one.

Observing the Graph:

• The graph appears to have three distinct x-intercepts: one at $x = 0$, and two others at points $x = a$ and $x = b$, where $a$ and $b$ are different from 0.
• It has a double root at both $x = a$ and $x = b$, meaning the graph touches the x-axis at these points without crossing it. This suggests that the factors $(x - a)$ and $(x - b)$ should be squared.
• Additionally, the graph crosses the x-axis at $x = 0$, implying $x$ is a factor.
• The shape of the graph shows that it behaves like a cubic function at $x = 0$, as it crosses the x-axis rather sharply, indicating an odd multiplicity.

Evaluating the Options:

1. $f(x) = x(x - a)^3(x - b)^3$: This would have the graph crossing the x-axis at $x = a$ and $x = b$ due to the odd powers (3), but the graph only touches these points, not crosses them, so this is incorrect.

2. $f(x) = (x - a)^2(x - b)^4$: There is no factor for $x = 0$, so this is incorrect since the graph crosses the x-axis at $x = 0$.

3. $f(x) = x(x - a)^6(x - b)^2$: While this includes the $x = 0$ factor, the powers of 6 and 2 are too large for the behavior seen at $x = a$ and $x = b$, which would make the graph flatter than it appears.

4. $f(x) = x(x - a)^5(x - b)^2$: This fits the criteria:

   • $x$ is a factor, meaning the graph crosses the x-axis at $x = 0$.
   • The factor $(x - a)^5$ means the graph behaves like a cubic near $x = a$ but with an odd multiplicity (the graph crosses or changes direction), matching the behavior at $x = a$.
   • The factor $(x - b)^2$ gives a double root, which matches the graph touching but not crossing the x-axis at $x = b$.

Conclusion:

The correct function is $f(x) = x(x - a)^5(x - b)^2$, which is the last option.

Let me know if you need more details or have questions! Here are some related questions:

1. What is the significance of the multiplicity of roots in a polynomial function's graph?
2. How can we determine the degree of a polynomial from its graph?
3. What happens to the shape of the graph as the degree of a factor increases?
4. How do odd and even multiplicities affect how a graph crosses the x-axis?
5. Can a polynomial have multiple roots at the same x-value?

Tip: The multiplicity of a root directly affects how the graph behaves at that x-value—odd multiplicities result in the graph crossing the axis, while even multiplicities cause it to touch the axis but not cross.