Let's analyze the given graph and determine the polynomial equation in factored form.

Step 1: Identify the roots (x-intercepts)

From the graph, the function crosses the x-axis at approximately:

• $x = -3$
• $x = 0$
• $x = 2$

Thus, the polynomial must have factors corresponding to these roots: $(x + 3), (x - 0), (x - 2)$

Step 2: Determine the nature of each root

• The root at $x = -3$ appears to be a single root.
• The root at $x = 0$ appears to be a double root (tangent to the x-axis).
• The root at $x = 2$ appears to be a single root.

Thus, the polynomial can be written as: $y(x) = a(x + 3)(x^2)(x - 2)$

Step 3: Determine the leading coefficient $a$

Observing the graph, the function appears to have a negative leading coefficient (since the right end of the curve moves downward). So, we assume $a < 0$.

Approximating a point from the graph, such as $(1, -4)$, we substitute:

$-4 = a(1 + 3)(1^2)(1 - 2)$

$-4 = a(4)(1)(-1)$

$-4 = -4a$

$a = 1$

Final Equation

$y(x) = (x + 3)x^2(x - 2)$

Would you like a breakdown of the expansion or any clarifications?

Related Questions:

1. How do you determine the degree of the polynomial from its graph?
2. What happens if a root has an odd vs. even multiplicity?
3. How do you determine the leading coefficient using another given point?
4. What are inflection points, and how do they relate to polynomial functions?
5. How can you use synthetic division to verify the factors of a polynomial?

Tip:

When a polynomial touches the x-axis but does not cross it, the corresponding root has an even multiplicity (like $x^2$).