I am trying to find the y-intercepts without using a graphing calculator. How do I go about with that equation?

In order for a line to touch on the y axis it must be at a point (0, y) at any y point. So you substitue zero in for all x variables in the equation, so it would become y= (0^2-0-2)/ (0^2-16). Which then goes to y=(0-2)/(0-16). -> y= -2/-16 ->y=1/8 (answer).