To sketch the graph of the equation [tex]\( y = (x-1)^2 + 2 \)[/tex] and identify its axis of symmetry, let's follow these steps:
1. Identify the Form of the Equation:
- The equation [tex]\( y = (x-1)^2 + 2 \)[/tex] is in the standard form of a parabola, [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
2. Determine the Vertex:
- In the given equation, [tex]\( h = 1 \)[/tex] and [tex]\( k = 2 \)[/tex]. Therefore, the vertex of the parabola is at the point [tex]\((1, 2)\)[/tex].
3. Determine the Axis of Symmetry:
- The axis of symmetry for a parabola in the form [tex]\( y = a(x - h)^2 + k \)[/tex] is the vertical line [tex]\( x = h \)[/tex]. In this case, [tex]\( h = 1 \)[/tex].
- Thus, the axis of symmetry is [tex]\( x = 1 \)[/tex].
4. Sketch the Graph:
- Plot the vertex [tex]\((1, 2)\)[/tex] on the coordinate plane.
- Since the coefficient [tex]\( a \)[/tex] in [tex]\( y = a(x - h)^2 + k \)[/tex] is positive, the parabola opens upwards.
- Draw a symmetric parabola with its vertex at [tex]\((1, 2)\)[/tex] and opening upwards.
5. Ensure All Details are Included:
- The vertex [tex]\((1, 2)\)[/tex] is the highest or lowest point on the graph, namely the lowest because the parabola opens upwards.
- The parabola is symmetric about the vertical line [tex]\( x = 1 \)[/tex], i.e., for every point [tex]\( (x, y) \)[/tex] on one side of the axis of symmetry there is a corresponding point [tex]\( (2 - x, y) \)[/tex] on the other side.
6. Identify the Correct Option:
- Given the choices for the axis of symmetry are [tex]\( x = 1 \)[/tex], [tex]\( x = 2 \)[/tex], [tex]\( x = -1 \)[/tex], and [tex]\( x = -2 \)[/tex]:
- As we have determined, the correct axis of symmetry is [tex]\( x = 1 \)[/tex].
Therefore, the axis of symmetry for the graph of [tex]\( y = (x-1)^2 + 2 \)[/tex] is [tex]\( x = 1 \)[/tex].
