https://file.aiquickdraw.com/mathbot/file/17273668715975gqluwmg.jpg

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?

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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!

Use the points on the graph to state a specific equation for the graphed function: g(x) = (1/b(x - h))^3 + k

Learn how to find the specific equation of a cubic function using points from its graph. This problem uses the form g(x) = (1/b)(x - h)^3 + k and explores transformations, inflection points, and cubic function properties. Suitable for Grades 10-12.

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