To determine which function could represent the graph, let’s analyze the key characteristics of the graph, especially the zeroes, multiplicities, and general behavior.

Observations from the Graph

1. Roots: The graph crosses the x-axis at least three distinct points, indicating at least three distinct roots.
2. Multiplicity of Roots: The graph shows some roots with an odd multiplicity (it crosses the x-axis) and some with an even multiplicity (it touches the x-axis without crossing it).
3. Degree of Polynomial: The overall shape suggests a higher degree polynomial with positive leading behavior (as $x \to \infty$, $f(x) \to \infty$).

Analyzing Each Option

Let’s break down each given function to match these characteristics:

1. Option 1: $f(x) = x^2(x - a)^2(x - b)^4(x - c)$

   • Degree: $2 + 2 + 4 + 1 = 9$
   • Roots: $x = 0, x = a, x = b, x = c$
   • The graph of a degree-9 polynomial would generally end with opposite signs as $x \to \pm \infty$. However, this graph ends with both ends up, so this option may not fit.

2. Option 2: $f(x) = x^3(x - a)^3(x - b)(x - c)^2$

   • Degree: $3 + 3 + 1 + 2 = 9$
   • Roots: $x = 0, x = a, x = b, x = c$
   • Similar to Option 1, this has a degree-9, but with a similar issue of sign behavior at both ends.

3. Option 3: $f(x) = x^4(x - a)(x - b)^3(x - c)^3$

   • Degree: $4 + 1 + 3 + 3 = 11$
   • Roots: $x = 0, x = a, x = b, x = c$
   • Degree-11 would also show opposite end behavior, which doesn’t match.

4. Option 4: $f(x) = (x - a)^2(x - b)(x - c)^6$

   • Degree: $2 + 1 + 6 = 9$
   • Roots: $x = a, x = b, x = c$ only, no $x = 0$.
   • Matches an odd multiplicity for root