Question #08e86

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Answer 1 · 2017-11-20T09:11:20

$Y = (\frac{1}{5}) {(x + 3)}^{2}$ $Y = (1/5)(x + 3) ^ 2$ 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 $(- 3, 0)$ $(-3, 0)$

Directrix is at $Y = (- 1.25)$ $Y = (-1.25)$

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 $P = + 1.25$ $P = + 1.25$

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

${(x - h)}^{2} = 4 p (y - k)$ $(x - h)^2 = 4p(y - k)$ ( Formula )

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

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

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,

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

${(x + 3)}^{2} = 5 (y - 0)$ $( x + 3 ) ^ 2 = 5 ( y - 0 )$

${(x + 3)}^{2} = 5 y$ $( x + 3 ) ^ 2 = 5y$

Hence, we get,

$y = {(\frac{1}{5}) {(x + 3)}^{2}}$ $y = { ( 1 / 5 ) ( x + 3 ) ^ 2}$

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