To find the equation of the parabola with the given vertex and focus, we start with the vertex \((1, 3)\) and the focus \(\left(\frac{1}{2}, 3\right)\).
Step 1: Determine the orientation of the parabola
The vertex is at \((1, 3)\) and the focus is at \(\left(\frac{1}{2}, 3\right)\). Since the focus is to the left of the vertex, the parabola opens to the left.
Step 2: Use the vertex form of the parabola equation
The vertex form of a parabola that opens to the left is given by:
\[ (y - k)^2 = -4p(x - h) \]
where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus.
Step 3: Calculate \(p\)
In this case:
- Vertex \((h, k) = (1, 3)\)
- Focus \(\left(\frac{1}{2}, 3\right)\)
The distance \(p\) is the horizontal distance from the vertex to the focus:
\[ p = 1 - \frac{1}{2} = \frac{1}{2} \]
Since the parabola opens to the left, \(p\) will be negative, so:
\[ p = -\frac{1}{2} \]
Step 4: Plug into the equation
Now substituting \(h = 1\), \(k = 3\), and \(p = -\frac{1}{2}\) into the equation:
\[ (y - 3)^2 = -4 \left(-\frac{1}{2}\right)(x - 1) \] \[ (y - 3)^2 = 2(x - 1) \]
Final Equation
Rearranging gives:
\[ (y - 3)^2 = 2(x - 1) \]
Thus, the correct equation of the parabola is:
\[ \boxed{(y - 3)^2 = 2(x - 1)} \]
So the final answer is the second option:
(y - 3)^2 = 2(x - 1).