sketch the graph of each function and spec-
ify all x- and y-intercepts. Y=-(x-1)^4

Let's go through the steps to sketch the graph of the function \( Y = -(x - 1)^4 \) and find the x- and y-intercepts.

### 1. Identify the basic shape of the function:
- The given function \( Y = -(x - 1)^4 \) is a quartic function, where:
  - The negative sign in front of the expression indicates that the graph is flipped upside down.
  - The function \( (x - 1)^4 \) is symmetric about the line \( x = 1 \) and has a "W" shape when positive, but since it's negative, it will be an inverted "W" or a "U" shape with the ends pointing downward.

### 2. Vertex of the graph:
- The vertex occurs where the inside of the function is zero, i.e., \( x = 1 \).
- Substituting \( x = 1 \) into the equation gives:
  \[
  Y = -(1 - 1)^4 = -0 = 0
  \]
  So, the vertex is at \((1, 0)\).

### 3. X-intercepts:
- To find the x-intercepts, set \( Y = 0 \) and solve for \( x \):
  \[
  0 = -(x - 1)^4
  \]
  This equation is true when \( (x - 1)^4 = 0 \), which implies \( x = 1 \).
  - Therefore, there is a single x-intercept at \( x = 1 \).

### 4. Y-intercept:
- To find the y-intercept, set \( x = 0 \) and solve for \( Y \):
  \[
  Y = -(0 - 1)^4 = -1
  \]
  So, the y-intercept is at the point \((0, -1)\).

### 5. Sketch the graph:
- The graph is symmetric about \( x = 1 \), has a vertex at \((1, 0)\), and opens downwards.
- The x-intercept is at \((1, 0)\), and the y-intercept is at \((0, -1)\).

### Summary:
- **Vertex**: \((1, 0)\)
- **X-intercept**: \((1, 0)\)
- **Y-intercept**: \((0, -1)\)

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Learn how to sketch the graph of Y = -(x - 1)^4 and find its x- and y-intercepts step-by-step. This problem covers graphing quartic functions and understanding symmetry in functions. Suitable for high school students.

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