SL 2.8—Reciprocal and rational functions

Use asymptotes and intercepts to sketch rational functions.

For a rational function, excluded denominator zeros create vertical asymptotes; end behaviour gives horizontal or oblique asymptotes.

Worked example
f(x)=1/(x−2)+3 has vertical asymptote x=2 and horizontal asymptote y=3.

Worked example
Can the graph cross a vertical asymptote? no, the function is undefined there.

Common boundary
An asymptote is not always a line the graph never approaches closely.

Linear-over-linear example: for $f(x)=\frac{2x-3}{x+4}$, the vertical asymptote is $x=-4$ and the horizontal asymptote is $y=2$ (the ratio of leading coefficients). The x-intercept is $x=3/2$ and the y-intercept is $f(0)=-3/4$. Plot these features before sketching the branches; $x=-4$ is excluded from the domain.