To find the equation of the parabola with the focus at \((3, 1)\) and the directrix \(y = -3\), we can follow these steps:
-
Find the vertex: The vertex of a parabola is located halfway between the focus and the directrix.
The focus is at \((3, 1)\) and the directrix is \(y = -3\). To find the y-coordinate of the vertex:
\[ \text{Vertex y-coordinate} = \frac{\text{Focus y-coordinate} + \text{Directrix y-coordinate}}{2} = \frac{1 + (-3)}{2} = \frac{-2}{2} = -1 \]
Since the x-coordinate of the vertex is the same as the focus's x-coordinate, the vertex is at \((3, -1)\).
-
Determine the direction of the parabola: Since the focus is above the directrix, the parabola opens upward.
-
Use the vertex form of the parabola equation: The equation of a parabola that opens upward is given by:
\[ (x - h)^2 = 4p(y - k) \]
where \((h, k)\) is the vertex, and \(p\) is the distance from the vertex to the focus (or from the vertex to the directrix).
The distance \(p\) can be calculated as follows:
- The y-distance from the vertex at \((3, -1)\) to the focus at \((3, 1)\) is \(1 - (-1) = 2\).
- Thus, \(p = 2\).
-
Substituting into the vertex form: Since the vertex is \((3, -1)\) and \(p = 2\):
\[ (x - 3)^2 = 4 \cdot 2 (y + 1) \]
Simplifying this gives:
\[ (x - 3)^2 = 8(y + 1) \]
Expanding and rearranging gives:
\[ (x - 3)^2 = 8y + 8 \] \[ x^2 - 6x + 9 = 8y + 8 \] \[ 8y = x^2 - 6x + 1 \] \[ y = \frac{1}{8}x^2 - \frac{3}{4}x + \frac{1}{8} \]
Thus, the equation in standard form is:
\[ y = \frac{1}{8}x^2 - \frac{3}{4}x + \frac{1}{8} \]
The closest match from the options given is:
y = Start Fraction 1 over 8 End Fraction x squared minus Start Fraction 3 over 4 End Fraction x plus Start Fraction 1 over 8 End Fraction.