The vertex form of a parabola is given by the equation:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola. For your problem, the vertex is \((1, 1)\), so we have:
\[ y = a(x - 1)^2 + 1 \]
Next, we need to find the value of \(a\). We can use the y-intercept, which is given as \((0, 2)\). We substitute \(x = 0\) and \(y = 2\) into the equation:
\[ 2 = a(0 - 1)^2 + 1 \]
This simplifies to:
\[ 2 = a(1) + 1 \]
\[ 2 = a + 1 \]
Subtract 1 from both sides:
\[ a = 1 \]
Now we can substitute \(a\) back into the vertex form equation:
\[ y = 1(x - 1)^2 + 1 \]
This can be simplified to:
\[ y = (x - 1)^2 + 1 \]
Therefore, the equation of the parabola in vertex form is:
\[ \boxed{y = (x - 1)^2 + 1} \]