Shifting a Graph — Cheat sheet

CHEAT SHEET · FREE

Key method

For $g(x)=f(x-h)+k$ , translate every point by $(h,k)$ . A forbidden input $u$ of $f$ becomes the forbidden input $u+h$ of $g$ . Vertical shifts do not change the domain.

Example

Starting from $f(x)=1/x$ , the function $g(x)=1/(x-2)+3$ has vertical asymptote $x=2$ and horizontal asymptote $y=3$ . Its domain excludes $2$ and its range excludes $3$ . The point $(1,1)$ becomes $(3,4)$ .

Avoid this mistake

A shift inside a denominator is still an input shift. Solve the denominator-zero equation rather than inferring a sign visually.