If we work out y in terms of x we get y=2+/-15*sqrt(((x+2)/20)^2-1), which only exists when (x+2)^2=>400, i.e., x=>18 or x<=-22. The asymptotes must therefore, it seems, be associated with these values. At x=18 or -22, y=2. The line y=2 represents a definable limit of the function that is attainable at points (18,2) and (-22,2), whereas an asymptote is not attainable. If we write x in terms of y we get x=-2+/-20*sqrt(1+((y-2)/15)^2), we can see that the square root term can never be negative. As y gets bigger, however, the square root approaches (y-2)/15, so x approaches -2+/-20(y-2)/15 which is -2+/-4/3(y-2). Similarly, y approaches 2+/-15(x+2)/20, which is 2+/-3/4(x+2). That is, both x and y tend to infinity positively and negatively, and an asymptote cannot be defined, unless we take the positive and negative slopes, defined by +/-3/4 as x and y tend to infinity. The asymptotes then form an X where the centre of the X is the point (-2,2), because x=-2 is halfway between x=18 and -22. The asymptotes are therefore two straight lines with opposite slopes in the form y=ax+b. Both lines intersect at (-2,2) so we can substitute these values for x and y to find b in each case. When we do this we get b=7/2 and 1/2, giving us the equations: y=(3/4)x+7/2 and y=-(3/4)x+1/2 or 4y=3x+14 and 4y=2-3x.
[The shape of the curve is interesting. The line y=2 is a reflector, on the positive side a curve emerges from (18,2) and moves off to the right as x and y get larger and larger. Underneath the line is its reflection which cuts the x axis at about 18.177 and continues to the right reflecting the upper part of the curve. The two halves together form U shapes, lying on their sides, with outward curving arms, and with the centre of the U's on the line y=2. The y axis is also a reflector, and the two halves of the curve are reflected on the left with the same hyperbolic shape as on the right but in the opposite direction. The x axis is cut at x=-22.177.]