To determine which equation corresponds to the given graph, let's analyze the transformation of the function $y = x^3$ based on the options and the provided graph.

Upon inspecting the graph, we observe:

• The graph seems to shift vertically upward by 4 units.
• The function has a steeper curve, indicating a scaling transformation.

Now, let's check each equation from the options:

a) $f(x) = \frac{1}{2} x^3 + 4$: This suggests a vertical shift of +4 and a vertical compression (scaling by $\frac{1}{2}$).

b) $f(x) = (2x)^3 + 4$: This represents a vertical shift of +4 and a horizontal compression by a factor of 2 (since $2x$ is inside the cube).

c) $f(x) = \frac{1}{2} (x - 4)^3$: This suggests a shift horizontally to the right by 4 units and a vertical compression by a factor of $\frac{1}{2}$.

d) $f(x) = (2(x - 4))^3$: This suggests a shift horizontally to the right by 4 units and a horizontal compression by a factor of 2.

The graph shows a vertical shift and a steeper curve, which means the function undergoes a scaling effect on the vertical axis, but without a horizontal shift. The most suitable match is option (b): $f(x) = (2x)^3 + 4$, because it combines the vertical shift of 4 and the increased steepness due to the horizontal compression by a factor of 2.

Would you like further clarification on this, or have any additional questions? Here are some related questions:

1. How do horizontal and vertical shifts affect the graph of a function?
2. What is the effect of a coefficient outside the function, like $\frac{1}{2}$ or 2, on the graph?
3. How do you identify a horizontal compression or expansion by looking at the graph?
4. What is the general transformation rule for cubic functions?
5. How would the graph change if the cube was $x^3 - 4$ instead of $(x - 4)^3$?

Tip: To identify transformations, look for vertical shifts (added constants) and changes in the steepness (caused by multiplying by a constant inside or outside the function).