Let's go through each part step-by-step:

(a) f(x) = x³, g(x) = (x − 3)³

The function g(x) = (x − 3)³ is derived from f(x) = x³ through a horizontal shift. Specifically:

• The equation g(x) = (x − 3)³ represents a shift of 3 units to the right compared to f(x) = x³.

How to get this graph:

• The graph of g(x) = (x − 3)³ is the graph of f(x) = x³ moved to the right by 3 units. This transformation does not affect the shape of the graph or the vertical stretching.

Thus, the correct transformation is:

• Shift right 3 units.

(b) f(x) = x³, g(x) = x³ − 3

The function g(x) = x³ − 3 is obtained from f(x) = x³ by a vertical shift.

• The equation g(x) = x³ − 3 represents a shift of 3 units downward compared to f(x) = x³.

How to get this graph:

• The graph of g(x) = x³ − 3 is the graph of f(x) = x³ moved down by 3 units. This transformation does not affect the shape of the graph, just the vertical position.

Thus, the correct transformation is:

• Shift down 3 units.

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Summary:

• (a) f(x) = x³ → g(x) = (x − 3)³: Shift right 3 units.
• (b) f(x) = x³ → g(x) = x³ − 3: Shift down 3 units.

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Would you like further clarification or more details on these transformations? Here are some related questions for further exploration:

1. How would the graph change if g(x) = (x + 3)³?
2. What happens if we stretch the graph vertically by a factor of 2 in both cases?
3. What is the effect of shifting f(x) = x³ up by 5 units instead of down by 3 units?
4. How would the graph of g(x) = (x − 3)³ look if we also stretch it vertically by a factor of 2?
5. How can we express the transformation for the graph of f(x) = x³ when shifting 3 units left and up by 3 units?

Tip: To quickly identify horizontal shifts in cubic functions, remember that the transformation (x − h)³ shifts the graph h units to the right, and (x + h)³ shifts it h units to the left.