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Question
Find the equation of the parabola with vertex left-parenthesis 1 comma 3 right-parenthesis and focus enlarged-left-parenthesis Start Fraction 1 over 2 End Fraction comma 3 enlarged-right-parenthesis.
(1 point)
Responses

left-parenthesis y minus 3 right-parenthesis squared equals negative 2 left-parenthesis x minus 1 right-parenthesis
Image with alt text: left-parenthesis y minus 3 right-parenthesis squared equals negative 2 left-parenthesis x minus 1 right-parenthesis

left-parenthesis y minus 3 right-parenthesis squared equals 2 left-parenthesis x minus 1 right-parenthesis
Image with alt text: left-parenthesis y minus 3 right-parenthesis squared equals 2 left-parenthesis x minus 1 right-parenthesis

left-parenthesis x minus 1 right-parenthesis squared equals negative 2 left-parenthesis y minus 3 right-parenthesis
Image with alt text: left-parenthesis x minus 1 right-parenthesis squared equals negative 2 left-parenthesis y minus 3 right-parenthesis

left-parenthesis x minus 1 right-parenthesis squared equals 2 left-parenthesis y minus 3 right-parenthesis

Answer 1

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)**.