Analyzing f(x) = (2x + 3)/(x - 1)

Algebra 2 · Axiom Academy

Work through domain, both asymptotes, both intercepts, and the behavior near the pole — the full feature checklist for a rational function. Fully analyze the rational function : find its domain, its vertical and horizontal asymptotes, its x - and y -intercepts, and its behavior near the pole — then sketch it. Vertical asymptote (dashed red) at x = 1; horizontal asymptote (dashed green) at y = 2; intercepts marked in orange. Both branches approach but never touch either asymptote. Nice work — you analyzed feature by feature, then read the complete picture off the graph. Domain: all real numbers except x = 1 , where the denominator is zero. Vertical asymptote: x = 1 — genuine, since the numerator is 5 there and does not also vanish. Horizontal asymptote: y = 2 — numerator and denominator share the same degree, so it's the ratio of leading coefficients, . Intercepts: y -intercept (0,-3) from f(0) ; x -intercept (-1.5,0) from 2x+3=0 . Behavior at the pole: as and as — opposite one-sided limits, exactly what a genuine vertical asymptote looks like. This same checklist — domain, asymptotes, intercepts, and behavior near any pole — analyzes any rational function before you ever plot a point.

This is the written version of the interactive lesson above. See the full Algebra 2 course.