Question

1. write the equation of the parabola.

$(x-3)^2=12(y-1)$

Explanation:

Step1: Identify vertex and direction

The vertex of the parabola is at $(3, 1)$, and it opens downward, so we use the vertex form for a vertical parabola: $(x-h)^2 = 4p(y-k)$, where $(h,k)$ is the vertex.

Step2: Find the value of $p$

The directrix (the horizontal dashed line) is at $y=4$. The distance from the vertex $(3,1)$ to the directrix is $p = 1 - 4 = -3$ (negative because the parabola opens downward).

Step3: Substitute values into the formula

Substitute $h=3$, $k=1$, and $p=-3$ into the vertex form:
$$(x-3)^2 = 4(-3)(y-1)$$
Simplify the right-hand side:
$$(x-3)^2 = -12(y-1)$$

Answer:

$\boldsymbol{(x-3)^2 = -12(y-1)}$