sketch the graph of each function and spec-
ify all x- and y-intercepts. Y=(x-2)^2+1

To sketch the graph of the function \( Y = (x - 2)^2 + 1 \), let's go through the following steps:

### 1. Identify the basic shape of the function:
- The given function \( Y = (x - 2)^2 + 1 \) is a quadratic function in the standard form of a parabola \( Y = a(x - h)^2 + k \), where:
  - \( a = 1 \) (indicating that the parabola opens upwards)
  - \( h = 2 \) (shifts the parabola horizontally to the right by 2 units)
  - \( k = 1 \) (shifts the parabola vertically upwards by 1 unit)

### 2. Vertex of the parabola:
- The vertex of the parabola is at the point \((h, k)\).
- So, the vertex here is \((2, 1)\).

### 3. X-intercepts:
- To find the x-intercepts, set \( Y = 0 \) and solve for \( x \):
  \[
  0 = (x - 2)^2 + 1
  \]
  \[
  (x - 2)^2 = -1
  \]
  Since the square of a real number cannot be negative, there are **no real x-intercepts** for this function.

### 4. Y-intercept:
- To find the y-intercept, set \( x = 0 \) and solve for \( Y \):
  \[
  Y = (0 - 2)^2 + 1 = 4 + 1 = 5
  \]
  So, the y-intercept is at the point \((0, 5)\).

### 5. Sketch the graph:
- The graph is a parabola opening upwards with its vertex at \((2, 1)\).
- The parabola does not intersect the x-axis (no x-intercepts).
- It intersects the y-axis at \((0, 5)\).

### Summary:
- **Vertex**: \((2, 1)\)
- **X-intercepts**: None
- **Y-intercept**: \((0, 5)\)

Would you like me to sketch the graph or go into more details on this?

Learn how to sketch the graph of Y = (x - 2)^2 + 1, a quadratic function with details on its vertex, x-intercepts, and y-intercept. This problem involves the vertex form of a quadratic equation and its interpretation. Suitable for students in Grades 9-10.

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