The given equation is \( y = -(x - 1)^3 - 1 \). This is a cubic function that has been transformed.

### Analysis of the Function:

1. **Standard Form**: The standard form of a cubic function is \( y = a(x - h)^3 + k \). In this case:
   - \( a = -1 \) (which indicates the function is reflected across the x-axis)
   - \( h = 1 \) (the function is shifted 1 unit to the right)
   - \( k = -1 \) (the function is shifted 1 unit downward)

2. **Vertex (Turning Point)**: The vertex of this cubic function, also known as the point of inflection, occurs at \( (h, k) = (1, -1) \).

3. **Behavior**:
   - As \( x \) increases, \( y \) will decrease initially (due to the negative coefficient \( -1 \)) and then increase after the inflection point at \( x = 1 \).
   - As \( x \) decreases, \( y \) will also decrease.

4. **Domain and Range**:
   - **Domain**: The function is defined for all real numbers \( x \). Therefore, the domain is \( (-\infty, \infty) \).
   - **Range**: Since it's a cubic function, the range is also all real numbers, \( (-\infty, \infty) \).

5. **Symmetry**: This function is neither symmetric about the x-axis nor the y-axis, but it does exhibit point symmetry about the inflection point \( (1, -1) \).

Would you like to see a graph of this function, or do you have any specific questions about it?

### Related Questions:

1. How do transformations affect the graph of a cubic function?
2. What is the significance of the point of inflection in a cubic function?
3. How does the negative sign in front of a cubic function affect its shape?
4. Can a cubic function have more than one real root? How does this function relate to that?
5. How would the graph of this function differ if the equation was \( y = (x - 1)^3 - 1 \) instead?

**Tip**: When analyzing transformations, remember that changes inside the function (e.g., \( x - 1 \)) affect the graph horizontally, while changes outside the function (e.g., adding or subtracting a constant) affect the graph vertically.

Explore the analysis of the cubic function Y=-(x-1)^3-1. Understand its transformations, vertex, behavior, domain, range, and symmetry. Detailed insights suitable for high school students.

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