In exercises 5-16, sketch the graph of each function and specify all x- and y-intercepts. y=-(x-1)^3-1

Let's analyze the function \( y = -(x - 1)^3 - 1 \) and determine the x- and y-intercepts, then describe how to sketch its graph.

### Step 1: Identify the x- and y-intercepts

#### 1. **Y-intercept**:
The y-intercept occurs where \( x = 0 \).
- Substitute \( x = 0 \) into the equation:
  \[
  y = -((0 - 1)^3) - 1 = -(-1)^3 - 1 = -(-1) - 1 = 1 - 1 = 0
  \]
- **Y-intercept**: \( (0, 0) \)

#### 2. **X-intercept**:
The x-intercept occurs where \( y = 0 \).
- Set \( y = 0 \) and solve for \( x \):
  \[
  0 = -(x - 1)^3 - 1
  \]
  \[
  (x - 1)^3 = -1 + 1 = 0
  \]
  \[
  x - 1 = 0
  \]
  \[
  x = 1
  \]
- **X-intercept**: \( (1, -1) \)

### Step 2: Sketching the Graph

1. **Start with the basic cubic function**: \( y = x^3 \).
   - This is a standard cubic curve passing through the origin, rising to the right and falling to the left.

2. **Apply the transformation**:
   - **Horizontal Shift**: The graph is shifted 1 unit to the right because of \( (x - 1) \).
   - **Vertical Shift**: The graph is shifted 1 unit down because of the \( -1 \) outside the cubic term.
   - **Reflection**: The negative sign in front of the cubic term \( -(x - 1)^3 \) reflects the graph across the x-axis.

3. **Mark the key points**:
   - The **point of inflection** is at \( (1, -1) \), where the curve changes direction.
   - The graph passes through the **y-intercept** at \( (0, 0) \).

### Final Graph

- The curve starts from the lower left (as \( x \) approaches negative infinity), passes through the y-intercept at \( (0, 0) \), and then through the x-intercept at \( (1, -1) \), before continuing upward to the upper right.
- Due to the reflection, the graph falls as \( x \) increases, passes through the inflection point, and then rises again.

Would you like to see a plotted graph of this function?

### Related Questions:

1. How does the reflection affect the symmetry of the graph?
2. What are the general steps for sketching a cubic function with transformations?
3. Can you determine the behavior of the graph near the intercepts?
4. How would additional transformations (like vertical stretching) affect this graph?
5. What would the graph look like if the equation were \( y = (x - 1)^3 - 1 \) without the reflection?

**Tip**: When sketching graphs of transformed functions, always start with the base function and apply each transformation step by step.

Explore the graph of y = -(x - 1)^3 - 1 with detailed analysis and step-by-step sketching instructions. Learn about its x- and y-intercepts, transformations, and graph behavior. Suitable for high school students.

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