The problem asks what transformations are needed to graph the function $r(x) = (x - 5)^3 + 3$ from the standard cubic function $f(x) = x^3$.

Breakdown of Transformations:

1. Horizontal Shift: The term $(x - 5)$ indicates a horizontal shift. Since it's $x - 5$, the graph is shifted 5 units to the right.

2. Vertical Shift: The term $+3$ outside the cubic expression means the graph is shifted 3 units up.

Correct Answer:

The graph of $f(x) = x^3$ should be horizontally shifted 5 units to the right and vertically shifted 3 units up, which corresponds to option B.

Would you like a more detailed explanation of function transformations or have any other questions?

Here are 5 related questions to explore:

1. How do you identify horizontal shifts in transformed functions?
2. What does a vertical shift look like in a cubic function?
3. How would the graph change if the function was $r(x) = (x + 5)^3 - 3$?
4. What effect does multiplying a function by a negative number have?
5. Can you apply these transformations to quadratic functions?

Tip: Always check if transformations are inside the parentheses (affecting horizontal shifts) or outside (affecting vertical shifts).