Write the equation of the parabola with a vertex of (4, 3) and a focus of (4, -3)

Answer 1

the focus is below the vertex, which means the parabola is vertical and opens downward.

the general standard equation for a downward opening parabola is y-k=-a(x-h)² where (h,k) is the vertex and a=1/4f

the vertex for such a parabola is the maximum point

the equation is of the form y-3=-(1/4f)(x-4)² where f is the distance from the vertex to the focus or 6.

y-the y value of the vertex = -[1/4(6)](x-the x value of the vertex)²

y-3 = -(1/24)(x-4)²

The directrix is the same distance from the vertex as from the vertex to the focus. The focus is 6 from the vertex, below the vertex. The directrix is 6 above the vertex, or y=9. The vertex is the midpoint between the focus and the directrix.

A parabola is the locus of all points equidistant from the focus and directrix.

Pick any point (x,y) on the parabola, say (-8,-3). Its distance to the focus must equal its distance to the directrix. The focus is (4,-3). From (-8,-3) to (4,-3) is 12, 4-(-8), from (-8,-3) to y=9 is 12, 9-(-3)

(-8,-3) is on the parabola as seen by substituting it into the equation for the parabola

y-3=-(1/24)(x-4)²

-3-3=-(1/24)(-8-4)²

-6=-144/24=-6