Sure, let's go through the process step by step for sketching the graph of the equation [tex]\(y = (x-1)^2 + 2\)[/tex] and identifying the axis of symmetry.
### Step 1: Identify the vertex of the parabola
The given equation is of the form [tex]\(y = a(x-h)^2 + k\)[/tex], which represents a parabola. For the equation [tex]\(y = (x-1)^2 + 2\)[/tex]:
- The vertex [tex]\((h, k)\)[/tex] can be directly identified from the equation as [tex]\((1, 2)\)[/tex].
### Step 2: Determine the direction of the parabola
Since the coefficient of [tex]\((x-1)^2\)[/tex] is positive (equal to 1), the parabola opens upwards.
### Step 3: Identify the axis of symmetry
The axis of symmetry of a parabola given by the form [tex]\(y = a(x-h)^2 + k\)[/tex] is the vertical line [tex]\(x = h\)[/tex]. In this case:
- The axis of symmetry is [tex]\(x = 1\)[/tex].
### Step 4: Plot the vertex and the axis of symmetry
- Plot the vertex at the point [tex]\((1, 2)\)[/tex] on the coordinate plane.
- Draw a vertical line passing through [tex]\(x = 1\)[/tex] to represent the axis of symmetry.
### Step 5: Determine additional points on the parabola
To accurately sketch the parabola, you can find additional points by substituting values of [tex]\(x\)[/tex] around the vertex:
- For [tex]\(x = 0\)[/tex]:
[tex]\[ y = (0-1)^2 + 2 = 1 + 2 = 3 \][/tex]
So, [tex]\((0, 3)\)[/tex] is a point on the parabola.
- For [tex]\(x = 2\)[/tex]:
[tex]\[ y = (2-1)^2 + 2 = 1 + 2 = 3 \][/tex]
So, [tex]\((2, 3)\)[/tex] is another point on the parabola.
### Step 6: Sketch the parabola
- Plot the points [tex]\((0, 3)\)[/tex] and [tex]\((2, 3)\)[/tex].
- Draw a smooth curve through these points and the vertex, making sure that both sides of the parabola are symmetric about the axis [tex]\(x = 1\)[/tex].
### Final Sketch
1. Vertex: [tex]\((1, 2)\)[/tex]
2. Axis of symmetry: [tex]\(x = 1\)[/tex]
3. The points [tex]\((0, 3)\)[/tex] and [tex]\((2, 3)\)[/tex] help in shaping the parabola.
### Conclusion
Upon examining the given options for the axis of symmetry:
- [tex]\(x = 1 \)[/tex] is the correct axis of symmetry for the quadratic equation [tex]\(y = (x-1)^2 + 2\)[/tex].
So, the axis of symmetry is [tex]\( \boxed{x=1} \)[/tex].
