To determine which function could represent the given graph, let’s analyze the characteristics of the graph:

1. Number of Roots: The graph crosses the x-axis at two distinct points (approximately $x = -2$ and $x = 2$), indicating it has two distinct real roots.

2. Multiplicity of Roots: At these crossing points, the graph touches the x-axis and turns around, suggesting each root has a multiplicity of at least two (i.e., the function "bounces" off the axis at each root).

3. Degree of the Polynomial: The graph has three turning points, which suggests the degree of the polynomial function is likely even and greater than 2. Given the options, the degree of the polynomial appears to be 4, which would account for three turning points.

Now, let’s evaluate each function to see which one matches these observations:

• Option 1: $f(x) = x(x - a)(x - b)^2$

  • This function has three roots: one at $x = 0$ (simple root), one at $x = a$ (simple root), and one at $x = b$ (double root).
  • This doesn’t match our observation since it should have only two roots with multiplicity two each.

• Option 2: $f(x) = (x - a)(x - a)(x - b)^2$

  • This function also has three roots, with two distinct roots $x = a$ and $x = b$. However, $x = a$ has a double root, and $x = b$ has a double root as well.
  • This doesn’t fit either since we only see two points where the function "bounces."

• Option 3: $f(x) = x(x - a)^3(x - b)$

  • This function has four roots, including $x = 0$ (simple root), $x = a$ (triple root), and $x = b$ (simple root).
  • This doesn’t fit either, as it would have more crossing points than observed.

• Option 4: $f(x) = x^2(x - a)^2(x - b)^2$

  • This function has two roots with a multiplicity of 2 each, which matches the two points where the function bounces off the axis.
  • This matches the behavior of the graph, with two roots where the graph "bounces" at each crossing point.

Based on the analysis, Option 4 $f(x) = x^2(x - a)^2(x - b)^2$ is the correct answer, as it accurately reflects the behavior and degree of the polynomial in the graph.

Would you like further details or have any questions?

Related Questions

1. How can the degree of a polynomial function be determined from its graph?
2. What does the multiplicity of a root indicate about the behavior of the graph near that root?
3. How can you determine the number of turning points for a polynomial function based on its degree?
4. What role does symmetry play in identifying polynomial functions from graphs?
5. How can end behavior help in identifying the degree and leading coefficient of a polynomial function?

Tip

When analyzing polynomial graphs, focus on the roots, their multiplicities, and the end behavior to gain insight into the function's possible form.