To write the formula for the polynomial function of least degree based on the graph provided, we need to identify the key features of the graph:

1. Roots (x-intercepts): These occur where the graph crosses the x-axis.

   • The graph crosses the x-axis at $x = -2$, so this is a root of multiplicity 1.
   • The graph touches the x-axis at $x = 1$ without crossing it, indicating a root of multiplicity 2.

2. Leading term and end behavior:

   • The graph starts in the top-left quadrant and ends in the bottom-right quadrant, indicating a negative leading coefficient for the polynomial.

3. Degree of the polynomial:

   • Since the graph appears to change direction twice (indicating 3 turns), the polynomial should be of degree 3 or 4.
   • Considering the roots and their multiplicities, the degree is 3 (since there's one simple root and one double root).

Polynomial formula:

The function $f(x)$ can be written as: $f(x) = -a(x + 2)(x - 1)^2$ Where $a$ is a leading coefficient that we can determine by checking another point on the graph.

Finding the leading coefficient:

From the graph, we can use a point like $f(0) = 4$ to determine the value of $a$.

Substitute $x = 0$ and $f(0) = 4$ into the equation: $4 = -a(0 + 2)(0 - 1)^2$ $4 = -a(2)(1)$ $4 = -2a$ $a = -2$

Thus, the polynomial function is: $f(x) = 2(x + 2)(x - 1)^2$

Final Answer:

$f(x) = 2(x + 2)(x - 1)^2$

Let me know if you'd like further clarification or more details!

Here are 5 related questions to expand your understanding:

1. What happens to the graph of a polynomial when a root has higher multiplicity?
2. How can we determine the degree of a polynomial by looking at the graph?
3. How does the leading coefficient affect the end behavior of a polynomial function?
4. What is the significance of a root where the graph only touches but does not cross the x-axis?
5. How do changes in multiplicity of roots affect the shape of a polynomial graph?

Tip: Always check the points and end behavior of a polynomial graph when determining the equation to ensure accuracy in your answer!