Question

What is the focus of the parabola?

y=−14x2−2x−2

Answer 1

First, we need to transform the equation into its standard form (x - h)²=4p(y - k).
Using completing the square method:

y = -14x² - 2x - 2
y = -14(x² + 2x/14) - 2
y = -14(x² + 2x/14 + (2/28)²) -2 + (2/28)²
y = -14(x + 1/14)² - 391/196
-1/14(y + 391/196) = (x + 1/14)²

This is a vertical parabola and its focus (h, k + p) is (-1/14, -391/196 + 1/56) = (-1/14, -775/392).

Or (-0.071,-1.977).