Q. Which graph shows the axis of symmetry for the function \(f(x)=(x-2)^2+1\)?

Step 1 – Recognize the form

The function is in vertex form \( f(x)=a(x-h)^{2}+k \), whose vertex is \( (h,k) \).

Step 2 – Read off the vertex

Compare \( f(x)=(x-2)^{2}+1 \) with \( a(x-h)^{2}+k \). Here \( h=2 \) and \( k=1 \), so the vertex is \( (2,1) \).

Step 3 – Axis of symmetry

For a parabola in vertex form, the axis of symmetry is the vertical line \( x=h \). Therefore the axis of symmetry is \( x=2 \).

Step 4 – Verification

Check symmetric inputs about \( x=2 \), for example \( x=3 \) and \( x=1 \):
\[
f(3)=(3-2)^{2}+1=1+1=2
\]
\[
f(1)=(1-2)^{2}+1=1+1=2
\]
Both outputs match, confirming symmetry about \( x=2 \).

Answer

The axis of symmetry is the vertical line \( x=2 \).