Question

What is the equation of the parabola in vertex form with vertex (2,-4) and directrix y=-6​

Answer 1

Answer:

$y=\dfrac{1}{8}(x-2)^2-4$

Step-by-step explanation:

The directrix y=-6 is parallel to the x axes, so parabola has equation of the form

$(x-x_0)^2=2px(y-y_0),$

where $(x_0,y_0)$ is the vertex of parabola and p is parabola's parameter.

By the definition, $\frac{p}{2}$ is the distance from the vertex to the directrix, so

$\dfrac{p}{2}=2\Rightarrow p=4$

Hence, the equation of parabola is

$(x-2)^2=8(y+4)$

See attached diagram for the graph of parabola and its directrix.

In vertex form this equation is

$y=\dfrac{1}{8}(x-2)^2-4$