The graph of `y = (x + 3)^(3) + 1` is the graph of `y = x^(3)` shifted

Options

A. 3 units to the right and I unit down
B. 3 units to the left and I unit down
C. 3 units to the left and I unit up
D. 3 units to the right and I unit up

Answer

To solve the problem, we need to analyze how the function \( y = (x + 3)^3 + 1 \) is derived from the basic cubic function \( y = x^3 \).

### Step-by-Step Solution:

1. **Identify the base function**: The base function is \( y = x^3 \).

2. **Understand the transformation**: The expression \( (x + 3)^3 \) indicates a horizontal shift. Specifically, the term \( +3 \) inside the parentheses means that the graph of \( y = x^3 \) is shifted to the left by 3 units.

3. **Apply the vertical shift**: The \( +1 \) outside the cube indicates a vertical shift. This means that after shifting the graph left by 3 units, we then shift it up by 1 unit.

4. **Combine the transformations**: Therefore, the graph of \( y = (x + 3)^3 + 1 \) can be described as:
- Shift the graph of \( y = x^3 \) to the left by 3 units.
- Then, shift the resulting graph up by 1 unit.

5. **Conclusion**: The final transformation results in the graph of \( y = (x + 3)^3 + 1 \) being the graph of \( y = x^3 \) shifted left by 3 units and up by 1 unit.

### Final Answer:
The graph of \( y = (x + 3)^3 + 1 \) is the graph of \( y = x^3 \) shifted 3 units to the left and 1 unit up.

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