To find the specific equation of the polynomial function $g(x)$, we will use the general form given in the image:

$g(x) = \left(\frac{1}{b}(x - h)\right)^3 + k$

Step 1: Identify the turning point (h, k)

The turning point (inflection point) on the graph appears at $(1, 0)$. This gives us the values for $h$ and $k$, so:

• $h = 1$
• $k = 0$

Substituting $h$ and $k$ into the equation: $g(x) = \left(\frac{1}{b}(x - 1)\right)^3$

Step 2: Use another point to find $b$

Now, use one of the other points on the graph. Let's choose $(5, 1)$. Substituting $x = 5$ and $g(x) = 1$ into the equation: $1 = \left(\frac{1}{b}(5 - 1)\right)^3$ $1 = \left(\frac{1}{b} \cdot 4\right)^3$ $1 = \left(\frac{4}{b}\right)^3$ Now take the cube root of both sides: $1 = \frac{4}{b}$ So, $b = 4$.

Step 3: Final equation

Substitute $b = 4$ back into the equation: $g(x) = \left(\frac{1}{4}(x - 1)\right)^3$

Thus, the specific equation for the graphed function is: $g(x) = \left(\frac{1}{4}(x - 1)\right)^3$

Would you like more details or have any questions?

Here are 5 related questions:

1. How does changing the value of $b$ affect the shape of the graph?
2. What is the geometric significance of the point $(9, 2)$ on the graph?
3. How do you determine if a graph has an inflection point from its equation?
4. Can this method of finding $b$ be applied to other polynomial degrees (e.g., quadratics)?
5. How would the equation change if the graph were reflected over the x-axis?

Tip: Always use at least two points on the graph to confirm the accuracy of your function's equation!