To find the minimum value of the parabola given by the equation \( y = x^2 - 2x + 5 \), we can complete the square or use the vertex formula.
The standard form of a quadratic equation is \( y = ax^2 + bx + c \). Here, \( a = 1 \), \( b = -2 \), and \( c = 5 \).
The x-coordinate of the vertex, where the minimum value occurs (since \( a > 0 \)), can be found using the formula:
\[ x = -\frac{b}{2a} \]
Plugging in the values for \( b \) and \( a \):
\[ x = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1 \]
Now, we substitute \( x = 1 \) back into the equation to find the corresponding minimum \( y \) value:
\[ y = (1)^2 - 2(1) + 5 \] \[ y = 1 - 2 + 5 \] \[ y = 4 \]
Thus, the minimum value of the parabola \( y = x^2 - 2x + 5 \) is
\[ \boxed{4}. \]