none of this makes any sense ugh! anyone know??

y  = -14x^2 - 2x - 2

We want to get to the form     4p (y - k)  = ( x - h)^2  

Where (h, k)  is the vertex  and p is the distance from the vertex to the focus

We can factor this as

y =  -14 [x^2  + (1/7)x + (1/7) ]

Complete the square inside the brackets

Take (1/2)of (1/7)  = (1/14)....square this  = 1/196  ...add it and subtract it

So we have

y  =  -14 [ x^2 +(1/7)x + 1/196 +  (1/7) - (1/196) ]

Factor the first three terms  and simplify the last two.....so we have

y  =  -14 [  (x + 1/14)^2  + 27/196]      simplify

y = -14 ( x + 1/14)^2 - 27/14      add 27/14 to both sides

(y +27/14)  =  -14 ( x + 1/14)^2   multiply both sides by -1/14

(-1/14)(y + 27/14)  = ( x + 1/14)^2

Since  -1/14  =  4p....divide both sides by 4.....then  p  =  - 1/56

So...   the  vertex  is  (h, k) = ( -1/14, -27/14 )

And the focus  is given by ( h , k + p)  =  

( -1/14, - 27/14 - 1/56)  =

(- 1/14,  - 109/56 )

Here's the graph :  https://www.desmos.com/calculator/moljoeobcv

The equation of a parabola is y=18x^2−4x+41 .

What is the equation of the directrix?

We want the form  4p(y - k) = (x - h)^2      where (h, k) is the  vertex  and p is the distance between the vertex and the directrix

Factor out 18

y  =  18 ( x^2 - (2/9)x  +  41/18 )

Complete the square  inside the parentheses

Take (1/2) of (2/9) = (1/9) .....square it  = (1/81)....add and subtract it

y =  18 [  x^2 - (2/9)x + 1/81 +  41/18 - 1/81 ]   

factor the first three terms, simplify the last two

y  = 18 [ (x - 1/9)^2  + 367/162]    simplify

y  =  18 ( x - 1/9)^2 +  367/9     

subtract  367/9 from both sides

(y - 367/9) =  18 ( x - 1/9)^2      divide both sides by 18

(1/18) (y - 367/9)  = ( x - 1/9)^2

Since  4p  = 1/18,   divide both sides by 4....so p  =  1/72

The vertex  is  ( 1/9, 367/9)

And  the  equation of the directrix  is   y  = 367/9 - p    →  y  = 367/9 - 1/72  →

y = 2935 / 72

Here's the graph : https://www.desmos.com/calculator/qjwolpipur

The equation of a parabola is 132(y−2)^2 = x−1  .
What are the coordinates of the focus?

We want the  form

4p ( x - h)  = (y - k)^2     wher (h, k) is the vertex  and p  is the distance between the vertex and the focus

132 (y - 2)^2  =  (x - 1)     rearrange as

(x - 1)  =  132 ( y - 2)^2      divide both sides by 132

(1/132) ( x - 1)  = ( y - 2)^2

4p  =  (1/132)     divide both sides by 4   and p  =  1/528

And the focus  is given by

( h + p , k)   =  ( 1 + 1/528 , 2)   =  ( 529/528, 2 )

Here's the graph : https://www.desmos.com/calculator/cqctvmc09d