Question

What is the focus of the parabola?

Answer 1

We have that
y = −14x² − 2x − 2

First, we need to transform the equation into its vertex form 
(x - h)²=4p(y - k)
Group terms that contain the same variable
y = (−14x² − 2x )− 2
Factor the leading coefficient 
y = -14*(x² + (2/14)x )− 2
Complete the square  Remember to balance the equation 
y = -14*(x² + (2/14)x +(2/28)²-(2/28)²)− 2
y = -14*(x² + (2/14)x +(2/28)²)− 2+14*(2/28)²
y = -14*(x² + (2/14)x +(2/28)²)− 2+56/784

Rewrite as perfect squares

y = -14*(x+(2/28))²− 1512/784------>(x+1/14)²=(-1/14)*(y+1512/784)

4p=-1/14------> p=-1/56

This is a vertical parabola and its focus (h, k + p)

h=-1/14

k+p=(-1512/784)+(-1/56)----> (-1512-14)/784)----> -1526/784

the focus is

(-1/14,-1526/784)