Question

Question #5711b

Answer 1

`y=1/12(x-3)^2+2`

Explanation:

We can tell from the coordinates of the focus that it is above the Directrix. This mean the Parabola is opening up. If you draw a straight line down from the focus to the Directrix you will find the vertex sitting on that line half way down from the focus, i.e. the distance between the vertex and the focus is the same as the distance between the vertex and the Directrix.

This allows us to determine the coordinates of the vertex which is:

`vertex(2,-1)` note: -1 is 3 units down from focus and Directrix is 6 units down from focus.

Now let's find the vertex and focus coordinates of the translated parabola. It should be 1 unit to the right and 3 units up. i.e.:

`vertex(3,2)`

`focus(3,5)`

The equation of a regular Parabola is:

`y=a(x-h)^2+k`

where h and k are the x and y of the vertex.

So, the equation of our parabola becomes:

`y=a(x-3)^2+2`

Now we need to find the value of `a`.

The distance between focus and vertex is called `p`. It is 3 units and it is positive because the Parabola is opening up, i.e:

`p=3`

`a=1/(4p)`

`a=1/(4*3)=1/12`

Therefore, the equation of our translated Parabola is:

`y=1/12(x-3)^2+2`

Answer 2

`y = 1/12x^2 - 1/2x + 11/4`

Explanation:

Shift the focus 1 unit to the right and 3 units up:

`(2+1, 2+3) = (3,5)`

Shift the directrix 3 units up:

`y = -4+3`
`y = -1`

The focus and directrix for the second parabola are `(3,5)` and `y = -1`.

The distance between any point `(x,y)` on the parabola and the new focus is:

`d = sqrt((x-3)^2+(y-5)^2)`

The distance between any point on the parabola and the new directrix is:

`d = y-(-1)`

`d = y+1`

The definition of a parabola requires that these two distances are equal:

`y + 1 = sqrt((x-3)^2+(y-5)^2)`

Square both sides:

`(y + 1)^2 = (x-3)^2+(y-5)^2`

Expand the squares:

`y^2 + 2y+1 = x^2-6x+9+y^2-10y+25`

Combine like terms:

`12y = x^2 - 6x + 33`

Divide both sides by 12:

`y = 1/12x^2 - 1/2x + 11/4`