To solve the problem based on the polynomial function $g(x) = \frac{1}{6}(x - h)^3 + k$, you need to determine the values of $h$ and $k$ using the reference points from the graph. Here's a step-by-step outline:

1. Understand the General Form: The equation $g(x) = \frac{1}{6}(x - h)^3 + k$ represents a cubic function. The parameters $h$ and $k$ indicate the horizontal and vertical shifts, respectively.

2. Identify Reference Points: From the graph:

   • The point $(h, k)$ is the inflection point (center of symmetry).
   • Use the coordinates of this inflection point to directly determine $h$ and $k$.

3. Check Reference Points: Other points, such as $(1.5, 0)$, $(0, 1)$, and $(-1, -1)$, help verify the scaling factor and confirm the values of $h$ and $k$.

4. Use the Points to Verify: Substitute the coordinates of the points into the equation to verify:

   • For example, substituting $x = 1.5$ into the equation should give $g(x) = 0$.

Would you like me to analyze the specific values from the image and provide a detailed equation?