NapkinCalc

Precalculus B — Polynomials & Rational Functions

Rational functions & asymptotes

continues from lesson 1 — values defined earlier in the course stay live here

ELI5: divide one polynomial by another and the graph sprouts asymptotes — invisible lines it races toward but never touches. Where the denominator hits zero, the function blows up: a vertical asymptote. For (x² − 1)/(x − 2) that's x = 2.

(x^2 - 1) / (x - 2)
-50510-50510

vertical asymptote at x = 2 — undefined there, exploding on either side

Real-world hook: rational functions model diminishing returns and saturation — concentration vs dose, speed vs effort, lens focus distance.

Try it yourself: the function (x + 1)/(x² − 9) blows up where its denominator is zero. Enter the positive such x.

vayou=3va_{you} = 3 ✏️ Your turn: find the positive x where x² − 9 = 0 (a vertical asymptote). The check confirms x² = 9 without naming x.
✓ pass abs(vayou2−9)<1e−9andvayou>0abs(va_{you}^2 - 9) < 1e-9 and va_{you} > 0 green when your x makes the denominator zero