I'm guessing that you might be referring to the asymptote lines. If so:
y2/4-x2/2=0 gives the lines: y/2=±x/√2,
y=x√2 and y=-x√2 are the lines.
But the horizontal lines y=2 and y=-2 are true tangents at the minimum and maximum points (vertices).
Differentiating:
½ydy/dx-x=0. When the gradient dy/dx=0 (horizontal), x=0 so y2/4=1, y=±√4=±2. Extrema at (0,2) and (0,-2).