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. y = □(x □)^2□

Explanation:

Step1: Identify focus coordinates

From the graph, the focus is at the point $(-3, - 2)$.

Step2: Identify directrix equation

The directrix is a horizontal line. From the graph, the equation of the directrix is $y=-4$.

Step3: Find the value of $a$

The distance $p$ between the focus $(h,k + p)$ and the directrix $y=k - p$ is given by the formula $p$ is the distance between the focus and the directrix. Here, $p=\frac{-2-(-4)}{2}=1$. For a parabola of the form $y=a(x - h)^2+k$, and $p=\frac{1}{4a}$, so $a=\frac{1}{4p}$. Since $p = 1$, $a=\frac{1}{4}$. The vertex of the parabola is the mid - point between the focus and the point on the parabola directly above/below the focus. The vertex is $(-3,-3)$.

Step4: Write the equation of the parabola

The standard form of a parabola with a vertical axis of symmetry is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex. Here, $h=-3,k = - 3,a=\frac{1}{4}$. So the equation of the parabola is $y=\frac{1}{4}(x + 3)^2-3$.

Answer:

Focus: $(-3,-2)$
Directrix: $y=-4$
$y=\frac{1}{4}(x + 3)^2-3$