Question

determine the coordinates of the focus and the equation of the directrix. then, complete the equation of the parabola. enter your answers in the boxes. be sure to include the correct signs.

Explanation:

Step1: Identify focus coordinates from graph

From the graph, the focus is at the point $(3,3)$.

Step2: Identify directrix equation from graph

The directrix is a horizontal line at $y = - 3$.

Step3: Recall parabola formula

The standard - form equation of a parabola with a vertical axis of symmetry is $(x - h)^2=4p(y - k)$, 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 vertex is the mid - point between the focus and the directrix. The $x$ - coordinate of the vertex is $x = 3$, and the $y$ - coordinate of the vertex is $\frac{3+( - 3)}{2}=0$. So the vertex is $(3,0)$. The distance $p$ from the vertex $(3,0)$ to the focus $(3,3)$ is $p = 3$.

Step4: Write parabola equation

Substitute $h = 3$, $k = 0$, and $p = 3$ into the formula $(x - h)^2=4p(y - k)$. We get $(x - 3)^2=12y$.

Answer:

Focus coordinates: $(3,3)$
Directrix equation: $y=-3$
Parabola equation: $(x - 3)^2=12y$