The function given is:

$f(x) = 0.003 \cdot (x - 1)^3 \cdot (x + 2)^2 \cdot (x - 4) \cdot (x + 5)$

Let's analyze the characteristics of this function, including its roots, end behavior, and general shape.

1. Roots of the Function:

The function has factors of the form $(x - a)^n$, where each factor gives a root $x = a$ with multiplicity $n$. The roots of $f(x)$ are:

• $x = 1$, with multiplicity 3.
• $x = -2$, with multiplicity 2.
• $x = 4$, with multiplicity 1.
• $x = -5$, with multiplicity 1.

The multiplicity of each root affects how the graph behaves at that root:

• Odd multiplicities (like 1 and 3) imply that the function will cross the x-axis at those roots.
• Even multiplicities (like 2) imply that the function will touch the x-axis but not cross it.

2. End Behavior:

The degree of $f(x)$ is the sum of the multiplicities of each factor: $3 + 2 + 1 + 1 = 7$ Since the degree is 7 (an odd number) and the leading coefficient is $0.003$, a positive number, the end behavior is:

• As $x \to -\infty$, $f(x) \to -\infty$.
• As $x \to \infty$, $f(x) \to \infty$.

3. Shape of the Graph Near Each Root:

• At $x = 1$: Since it has an odd multiplicity of 3, the function crosses the x-axis and has a cubic-like behavior near this point.
• At $x = -2$: With an even multiplicity of 2, the function touches the x-axis and has a parabolic shape near this root.
• At $x = 4$ and $x = -5$: Both have an odd multiplicity of 1, so the function crosses the x-axis linearly at these points.

4. Scaling Factor:

The $0.003$ factor scales the function vertically, making the graph "flatter" compared to the unscaled polynomial. This impacts the steepness but does not change the locations of the roots or the general end behavior.

Summary:

The function has roots at $x = 1$, $x = -2$, $x = 4$, and $x = -5$, with respective multiplicities influencing the behavior at each point. The overall shape of the graph will show crossing at $x = 1$, $x = 4$, and $x = -5$, touching at $x = -2$, and growing positively as $x$ goes to infinity.

Would you like further details on graphing this function, analyzing specific points, or understanding transformations?

Related Questions

1. How does the multiplicity of each root affect the graph's appearance at that point?
2. What is the significance of the degree in determining the end behavior of a polynomial?
3. How does a scaling factor impact the graph of a polynomial function?
4. Why does an even multiplicity cause the function to touch the x-axis rather than cross it?
5. How would adding or changing a term in this function affect its roots and graph?

Tip:

For polynomials, examining the multiplicity of each root provides insight into how the function interacts with the x-axis at each point.