What is the focus of the parabola? y=14x2-x-1

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This is a vertical parabola which opens upwards

we have the general formula

(x - h) ^2 = 4p (y - k) where p is the y coordinate of the focus

y/14 = x^2 - x / 14 - 1/14

add 1/14 to both sides

y/14 + 1 / 14 = x^2 - x / 14

now complete the square

y/14 + 1/14 = x^2 - x/14 + 1/28^2

y/14 + 1/14 = (x - 1/28) ^2

(x - 1/28) ^2 = 1/14 (y + 1)

comparing this with the general form:-

4p = 1/14

p = 1/56

so the focus is at (h, p) = (1/28, 1/56)