The given graph is a parabola opening upwards with vertex at (2,1). The standard form is $y=a{(x-h)}^2+k$. Here, h=2 and k=1. The graph passes through (0,5). Substituting gives 5 = a(0-2)² + 1 → 5 = 4a + 1 → a=1. Therefore, the equation is $y={(x-2)}^2+1$.

Final Answer: The first option, $y={(x-2)}^2+1$, is correct.