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We are asked to determine which graph corresponds to the equation \( y = 1 - x^3 \).

### Step-by-step analysis:

1. **Nature of the function \( y = 1 - x^3 \)**:
   - This is a cubic function, where the dominant term is \( -x^3 \).
   - Cubic functions have general shapes resembling an S-curve, but the sign of the leading coefficient determines the direction of the tails:
     - For \( x^3 \) (positive), the curve falls to the left and rises to the right.
     - For \( -x^3 \) (negative), the curve rises to the left and falls to the right.

2. **Critical points**:
   - At \( x = 0 \), \( y = 1 - 0^3 = 1 \). Thus, the curve passes through the point \( (0, 1) \).
   - The derivative of \( y = 1 - x^3 \) is \( \frac{dy}{dx} = -3x^2 \). The derivative is zero at \( x = 0 \), which indicates a horizontal tangent line at \( (0, 1) \).

3. **Behavior of the function as \( x \to \infty \) and \( x \to -\infty \)**:
   - As \( x \to \infty \), \( -x^3 \) dominates, making \( y \to -\infty \). Thus, the right tail goes down.
   - As \( x \to -\infty \), \( y \to \infty \) since \( -(-x)^3 \) leads to \( y \to \infty \). Thus, the left tail goes up.

4. **Graph matching**:
   - We expect the graph to have the following features:
     - A point at \( (0, 1) \) with a horizontal tangent.
     - As \( x \) increases, the curve goes down.
     - As \( x \) decreases, the curve goes up.

#### Analyzing the options:
- **Option A** shows a curve rising to the left and falling to the right, with a horizontal tangent at \( (0, 1) \), which matches the behavior of \( y = 1 - x^3 \).
- **Option B** has the curve flipped with incorrect end behavior (falls to the left and rises to the right).
- **Option C** shows a curve that does not match the cubic nature (incorrect shape).
- **Option D** also has incorrect end behavior.

### Conclusion:
The correct graph is **(A)**.

Would you like further details, or do you have any questions?

Here are 5 related questions:
1. What does the general shape of cubic functions depend on?
2. How can you determine the direction of a curve from its leading term?
3. What role do derivatives play in determining graph features like tangents?
4. How does the value of the function at \( x = 0 \) influence graph placement?
5. Can we determine concavity from the second derivative of a function?

**Tip**: For cubic functions, always pay attention to the sign of the leading term to understand the overall direction of the graph.

Learn how to graph the function y = 1 - x^3 by analyzing its shape, end behavior, and critical points. This problem covers concepts in algebra and calculus, including cubic functions and derivatives. Suitable for students in Grades 10-12.

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