Question

What is the equation of the quadratic graph with a focus of (1,3) and a directrix of y=1?

Answer 1

check the picture below.

so the directrix is below the focus point, meaning the parabola is opening upwards, meaning is also a vertical parabola, meaning as well that the "x" variable is the one that is squared.

bearing in mind that the vertex is half-way between the focus point and the directrix, as you see there, h,k will then be (1,2).

the distance from the vertex to either the directrix or focus point is "p", and as you see there, is 1 unit, and since the parabola is opening upwards, is a positive 1.

$\bf \textit{parabola vertex form with focus point distance} \\\\ \begin{array}{llll} 4p(x- h)=(y- k)^2 \\\\ \boxed{4p(y- k)=(x- h)^2} \end{array} \qquad \begin{array}{llll} vertex\ ( h, k)\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix} \end{array}\\\\ -------------------------------\\\\ \begin{cases} h=1\\ k=2\\ p=1 \end{cases}\implies 4(1)(y-2)=(x-1)^2\implies 4(y-2)=(x-1)^2 \\\\\\ y-2=\cfrac{(x-1)^2}{4}\implies y=\cfrac{1}{4}(x-1)^2+2$