Question

Question #08e86

Answer 1

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

Explanation:

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.