Sure, let's explore how to sketch the graph of the equation [tex]\( y = (x-1)^2 + 2 \)[/tex] and identify its axis of symmetry step-by-step.
1. Identify the form of the equation:
The given equation is [tex]\( y = (x-1)^2 + 2 \)[/tex]. This is in the vertex form of a quadratic equation, which can be expressed generally as [tex]\( y = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
2. Find the vertex:
From [tex]\( y = (x-1)^2 + 2 \)[/tex], we can see that:
- [tex]\( h = 1 \)[/tex]
- [tex]\( k = 2 \)[/tex]
Therefore, the vertex of the parabola is at [tex]\( (1, 2) \)[/tex].
3. Determine the axis of symmetry:
The axis of symmetry of a parabola in vertex form [tex]\( y = a(x-h)^2 + k \)[/tex] is the vertical line that passes through the vertex. This means [tex]\( x = h \)[/tex].
- Since [tex]\( h = 1 \)[/tex], the axis of symmetry is [tex]\( x = 1 \)[/tex].
4. Plot key points:
To sketch the graph, plot the vertex [tex]\((1, 2)\)[/tex] and a few additional points to get the shape of the parabola.
- For [tex]\( x=0 \)[/tex]:
[tex]\( y = (0-1)^2 + 2 \)[/tex]
[tex]\( y = 1 + 2 = 3 \)[/tex]
[tex]\((0, 3)\)[/tex]
- For [tex]\( x=2 \)[/tex]:
[tex]\( y = (2-1)^2 + 2 \)[/tex]
[tex]\( y = 1 + 2 = 3 \)[/tex]
[tex]\((2, 3)\)[/tex]
These are just a few points to help you sketch. Remember that the parabola will be symmetric about the line [tex]\( x = 1 \)[/tex].
5. Sketch the parabola:
- Draw the vertex [tex]\((1, 2)\)[/tex].
- Draw the points [tex]\((0, 3)\)[/tex] and [tex]\((2, 3)\)[/tex], which are symmetric about [tex]\( x = 1 \)[/tex].
- Draw a smooth curve through these points to form the parabola opening upwards.
6. Label the axis of symmetry:
- Draw a vertical dashed line through [tex]\( x = 1 \)[/tex], which is the axis of symmetry of the parabola.
So, the axis of symmetry for the graph of [tex]\( y = (x-1)^2 + 2 \)[/tex] is:
[tex]\[ x = 1 \][/tex]
Therefore, among the given options, the correct answer is:
[tex]\[ x = 1 \][/tex]
