The reflector of a satellite dish is in the shape of a parabola with a diameter of 4 feet and a depth of 2 feet. to get the maximum reception we need to place the antenna at the focus. write the equation of the parabola of the cross section of the dish, placing the vertex of the parabola at the origin. convert the equation into standard form, if necessary. what is the defining feature of the equation that tells us it is a parabola? describe the graph of the parabola. find the vertex, directrix, and focus. use the endpoints of the latus rectum to find the focal width. how far above the vertex should the receiving antenna be placed?

A parabola is a figure with one vertex, and extending in curved lines at both sides creating an opening. It is stated here that the vertex is placed at the origin, thus it is at (0,0). It is also said that the antenna is placed ABOVE the vertex, so the parabola is opening upwards. The general formula for this is:

(x-h)^2 = 4a(y-k), where the origin is (h,k) therefore h=0 and k=0; a is the distance of focus to the vertex. Let's use the value of the depth and the diameter to fully define the equation. Since the depth is 2 feet, the diameter is 4, and the origin is at (0,0), then the parabola must pass the coordinates (4,2) and (-4,2) situated at opposite sides of each other. Let's use one of these (any of these two will yield the same answer):

(4-0)^2 = 4a(2-0)
4a = 8
a = 2

The lactus rectum of the parabola in the equation is 4a. Therefore, the length of the lactus rectum is 8 units. Then, the antenna should be placed at the focus which is 2 units above the vertex. The standard form of the parabola would then be: x^2 = 8y. The defining feature of a parabola is when its equation only has either x^2 or y^2, but not both.

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