The image shows a graph of a function P. Observing the graph, it is a parabola opening upwards with its vertex at (2,1). The general form of a parabola is $y=a{(x-h)}^2+k$, where (h,k) is the vertex. Here, h=2 and k=1. The parabola passes through (0,5), so we substitute to find a: $5=a{(0-2)}^2+1$, which gives $5=4a+1$, so a=1. Therefore, the equation is $P\left(x\right)={(x-2)}^2+1$.

Final Answer: The third option.