To find the polynomial function that corresponds to this graph, let’s analyze its key characteristics.

Step 1: Observing the Graph's Roots and Degree

From the graph, the polynomial seems to have three real roots, located at approximately:

• $x = -3$
• $x = 0$
• $x = 2$

Since the graph appears to be a smooth curve with no sharp turns, we are likely dealing with a polynomial with real coefficients. Additionally, the graph has a local maximum and minimum, which suggests the polynomial is of degree three (cubic polynomial).

Step 2: Constructing the Polynomial in Factored Form

For a cubic polynomial, the function $f(x)$ can be written as: $f(x) = a(x + 3)(x)(x - 2)$ where $a$ is a constant that determines the vertical stretch or compression of the function.

Step 3: Using a Point to Determine $a$

The problem states that the point $(-1, -9)$ lies on the graph. We can substitute this point into our equation to find $a$.

1. Substitute $x = -1$ and $f(x) = -9$: $-9 = a(-1 + 3)(-1)(-1 - 2)$

2. Simplify each term: $-9 = a(2)(-1)(-3)$ $-9 = 6a$

3. Solve for $a$: $a = \frac{-9}{6} = -\frac{3}{2}$

Step 4: Final Polynomial Function

Substitute $a = -\frac{3}{2}$ back into the equation: $f(x) = -\frac{3}{2}(x + 3)(x)(x - 2)$

This is the polynomial function in factored form.

Would you like a detailed explanation of any step, or do you have further questions?

Additional Questions to Explore

1. How would the polynomial change if the leading coefficient $a$ were positive?
2. What effect does changing the roots of the polynomial have on the graph?
3. Can you derive the expanded form of this polynomial?
4. What would the graph look like if the polynomial were of degree 4 instead of 3?
5. How does the sign of $a$ influence the direction of the polynomial's end behavior?

Tip

When working with graphs of polynomials, analyzing the roots and the behavior at those roots can help you determine the degree and form of the polynomial function quickly.