Rational functions

A reciprocal in disguise: A function y = (ax + b)/(cx + d) is a transformed reciprocal — same two-branch hyperbola shape, but with its own asymptotes. The vertical asymptote is where the denominator = 0.

Domain excludes the vertical asymptote: The domain is all x except the value that makes the denominator zero (here x ≠ 4).

Denominator zero = forbidden x: Solve cx + d = 0 for the vertical asymptote; that same x is excluded from the domain.

One vertical asymptote here: An (ax + b)/(cx + d) function has exactly one vertical asymptote (one value makes the linear denominator zero).

Ratio of the leading coefficients: As x → ±∞, the +b and +d barely matter, so y → a/c. The horizontal asymptote is y = a/c — the ratio of the x-coefficients on top and bottom.

Why a/c?: Divide top and bottom by x: (2 + 1/x)/(1 − 4/x) → 2/1 as the 1/x terms vanish.

Numerator zero → x-intercept: The x-intercept is where the numerator = 0 (a fraction is zero only when its top is zero). The y-intercept is the value at x = 0 (b/d). Then sketch the two branches around the asymptotes.