To deduce the function from the provided graph, let's analyze the key characteristics visually observed:

1. Zeros (Roots): The graph touches the x-axis at approximately:

   • $x = -6$, $x = -3$, $x = 0$, $x = 3$, and $x = 6$.

2. Behavior near Zeros:

   • At $x = -6$ and $x = 0$, the graph "bounces off" the x-axis, which suggests these are double roots.
   • At $x = -3$, $x = 3$, and $x = 6$, the graph crosses the x-axis, implying single roots at these points.

3. End Behavior:

   • The graph tends to infinity as $x \to -\infty$ and $x \to +\infty$, suggesting the degree of the polynomial is even.

4. Local Minima and Maxima:

   • There are local maximums between $x = -6$ and $x = -3$ as well as between $x = 3$ and $x = 6$.
   • There's a local minimum near $x = 0$.

Form of the Function:

Given the observations, a good candidate for the function is a polynomial that fits the roots and multiplicities: $f(x) = a(x + 6)^2(x + 3)(x - 0)^2(x - 3)(x - 6)$ Where:

• The double roots at $x = -6$ and $x = 0$ are represented by the squares.
• The single roots at $x = -3$, $x = 3$, and $x = 6$ are linear factors.

To further refine the function, we would need to scale it with an appropriate constant $a$. The value of $a$ could be determined by checking a specific point on the graph, such as the maximum or minimum.

Do you want to proceed with finding $a$, or do you have any questions about the function?

5 Related Questions:

1. How do we calculate the leading coefficient $a$ in polynomial functions?
2. What role do multiplicities of roots play in graph behavior?
3. How can we determine the degree of a polynomial from its end behavior?
4. Why does the graph bounce off the axis at some roots and cross at others?
5. How can you find the exact turning points of a polynomial function?

Tip: When analyzing polynomial graphs, always check for symmetries, as these can give clues about root multiplicities and overall degree!