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 parabola orientation

The parabola opens down - ward. The standard form of a parabola opening down - ward is $(x - h)^2=4p(y - k)$, where $(h,k)$ is the vertex, the focus is $(h,k + p)$ and the directrix is $y=k - p$.

Step2: Locate the vertex

From the graph, the vertex of the parabola is at the point $(0,3)$. So, $h = 0$ and $k = 3$.

Step3: Determine the value of $p$

The distance between the vertex $(0,3)$ and the focus $(0,1)$ is $|3 - 1|=2$. Since the parabola opens down - ward, $p=- 2$.

Step4: Find the focus

The focus of a parabola $(x - h)^2 = 4p(y - k)$ with $h = 0,k = 3,p=-2$ is $(h,k + p)=(0,3+( - 2))=(0,1)$.

Step5: Find the directrix

The equation of the directrix is $y=k - p$. Substituting $k = 3$ and $p=-2$, we get $y=3-( - 2)=y = 5$.

Step6: Write the equation of the parabola

Substitute $h = 0,k = 3,p=-2$ into the standard form $(x - h)^2=4p(y - k)$. We have $(x - 0)^2=4\times(-2)(y - 3)$, which simplifies to $x^{2}=-8(y - 3)$ or $y=-\frac{1}{8}x^{2}+3$.

Answer:

Focus: $(0,1)$; Directrix: $y = 5$; Equation of parabola: $y=-\frac{1}{8}x^{2}+3$