Question #08e86

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Answer 1 · 2017-11-20T09:11:20

#Y = (1/5)(x + 3) ^ 2# in Vertex Form

We are given the Vertex and Directrix of a Parabola.

We need to find the Equation of the Parabola.

On observing the graph in the question, we note that

Vertex is at #(-3, 0)#

Directrix is at #Y = (-1.25)#

We must remember that "Focus (P)" and the Directrix are the same distance from the Vertex, but on the opposite directions.

Hence, from our Focus, value of #P = + 1.25#

Since the given Directrix is a horizontal line, the Axis of Symmetry of this Parabola is Vertical.

The Vertex Form of the Equation of a Parabola we use the formula

#(x - h)^2 = 4p(y - k)# ( Formula )

where the Focus is #(h, k + p)#

In our problem, Focus is at #(1.25, 0)#

Using the ( Formula ) we have written above, we can find the Equation of a Parabola.

"h" if our x-Coordinate of the Vertex and "k" is the y-coordinate of the the Vertex.

So we get,

#{ x - ( -3 ) } ^ 2 = 4 ( 1.25 ) (y - 0 )#

#( x + 3 ) ^ 2 = 5 ( y - 0 )#

# ( x + 3 ) ^ 2 = 5y #

Hence, we get,

#y = { ( 1 / 5 ) ( x + 3 ) ^ 2}#

which is our required Vertex Form of the Equation of the Parabola.