HyperbolaAsGraph

Hyperbola as graph of a function

Consider the function y = f(x) = (ax²+bx)/(cx+d). Its graph coincides with a hyperbola.
To see it write the equation equivalently:
x(ax+b) - y(cx+d) = x(x+(b/a)) - (c/a)y(x+d/c) = 0.
The equation represents a particular conic member of the family of conics generated by the two singular conics:
(i) c₁(x,y) = x(x+(b/a)) = 0 (two parallel lines: the y-axis and a parallel to it).
(ii) c₂(x,y) = y(x+(d/c)) = 0 (two orthogonal lines: the x-axis and an orthogonal to it).
The two degenerate conics c₁, c₂ are combined by the constant -(c/a) to define the graph of the function f(x) in implicit form:
g(x,y) = c₁(x,y) -(c/a)c₂(x,y) = 0 <==> ac₁(x,y) - cc₂(x,y) = 0.

This shows that the graphs are conics and relates the value of (c/a) with the particular member passing through a definite point, say Q(1,k). Calculating g(1,k)=0 we find
a/c = k*(1+d/c)/(1+b/a).
Obviously cx+d=0 is an asymptote and this shows that the conic is a hyperbola.
The other asymptote assumed to be in the form y=Ax+B, results by finding {A,B} through limits. A is found by taking the limit f(x)/x and then B is found by taking the limit f(x)-Ax. Thus, A=a/c and B=(bc-ad)/c²=(a/c)((b/a)-(d/c)).

The above formulas allow the determination of the asymptote y=Ax+B of the hyperbola, given the values of (-d/c), (-b/a) and k. In the formulas the symbols {dc, ba, ac} represent respectively the values {-d/c, -b/a, a/c}.

Hyperbola as graph of a function

See Also