To determine which graph shows the axis of symmetry for the function [tex]\( f(x) = (x-2)^2 + 1 \)[/tex], let’s go through the necessary details step-by-step.
1. Identify the form of the function: The function given is [tex]\( f(x) = (x-2)^2 + 1 \)[/tex]. This is a quadratic function presented in vertex form, which is generally written as [tex]\( f(x) = a(x-h)^2 + k \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( h = 2 \)[/tex], and [tex]\( k = 1 \)[/tex].
2. Vertex form and axis of symmetry: In the vertex form [tex]\( f(x) = a(x-h)^2 + k \)[/tex], the vertex of the parabola is at the point [tex]\( (h, k) \)[/tex]. For the given function, the vertex is at [tex]\( (2, 1) \)[/tex].
3. Vertical line and axis of symmetry: The axis of symmetry for a parabola in vertex form is the vertical line that passes through the vertex. The equation for this line is [tex]\( x = h \)[/tex].
4. Determine [tex]\( h \)[/tex]: In this case, [tex]\( h = 2 \)[/tex].
5. Equation of the axis of symmetry: Therefore, the axis of symmetry is the vertical line [tex]\( x = 2 \)[/tex].
Now, we need to match the correct graph that shows the axis of symmetry as [tex]\( x = 2 \)[/tex]:
- On a coordinate plane, a vertical dashed line at [tex]\( (-2, 0) \)[/tex] is parallel to the [tex]\( y \)[/tex]-axis. This does not match since we need [tex]\( x = 2 \)[/tex].
- On a coordinate plane, a vertical dashed line at [tex]\( (2, 0) \)[/tex] is parallel to the [tex]\( y \)[/tex]-axis. This matches since [tex]\( x = 2 \)[/tex].
- On a coordinate plane, a vertical dashed line at [tex]\( (-1, 0) \)[/tex] is parallel to the [tex]\( y \)[/tex]-axis. This does not match since we need [tex]\( x = 2 \)[/tex].
- On a coordinate plane, a vertical dashed line at [tex]\( (1, 0) \)[/tex] is parallel to the [tex]\( y \)[/tex]-axis. This does not match since we need [tex]\( x = 2 \)[/tex].
Therefore, the graph that shows the axis of symmetry for the function [tex]\( f(x) = (x-2)^2 + 1 \)[/tex] is:
On a coordinate plane, a vertical dashed line at [tex]\( (2, 0) \)[/tex] parallel to the [tex]\( y \)[/tex]-axis.
