Reciprocal function

A hyperbola in two branches: The reciprocal function y = 1/x is a hyperbola with two separate branches — one in the top-right, one in the bottom-left. As x grows, y shrinks toward 0; as x nears 0, y shoots off to ±∞.

It never touches the axes: 1/x is never 0 (no x-intercept) and is undefined at x = 0 (no y-intercept). The axes are its asymptotes.

Big in, small out: Large x → tiny y; tiny x → huge y. The two are reciprocals, so the curve hugs both axes.

x ≠ 0, y ≠ 0: For y = 1/x: the domain is x ≠ 0 (can't divide by zero) and the range is y ≠ 0 (the output is never 0). The asymptotes are the lines x = 0 (vertical) and y = 0 (horizontal).

Asymptotes are dashed guides: Draw the asymptotes first (here the two axes); the branches then hug them.

The asymptotes move with the shift: y = 1/(x − h) + k is y = 1/x slid h right and k up. So the vertical asymptote becomes x = h and the horizontal asymptote becomes y = k.

Sign of h: 1/(x − 3) shifts right 3 (vertical asymptote x = +3); 1/(x + 3) shifts left (x = −3).

Asymptotes, then the two branches: To sketch: draw the asymptotes (dashed), find any intercepts (set x = 0 for the y-intercept, set y = 0 for the x-intercept), then draw the two branches hugging the asymptotes.

Reciprocals have at most one of each intercept: A shifted reciprocal crosses each axis at most once (sometimes not at all if an asymptote is in the way).