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?