Sketch graphs of rational functions, identifying intercepts, vertical asymptotes, and horizontal or oblique asymptotes.

What this dot point is asking

You need to sketch graphs of rational functions, identifying their intercepts and their vertical, horizontal and oblique asymptotes.

Vertical asymptotes and intercepts

For $f(x) = \dfrac{P(x)}{Q(x)}$ with $P$ and $Q$ sharing no common factors:

• Vertical asymptotes occur at the zeros of $Q(x)$: the graph shoots to $\pm\infty$ there. Check the sign of $f$ on each side to see whether the branch goes up or down.
• $x$-intercepts occur where $P(x) = 0$ (provided $Q \ne 0$ there).
• $y$-intercept is $f(0)$, when defined.

If $P$ and $Q$ share a factor, that point is a hole (removable discontinuity), not an asymptote - cancel the common factor first.

Behaviour far from the origin

Finding an oblique asymptote

When the numerator degree is exactly one more than the denominator, polynomial long division writes $f(x) = (\text{linear}) + \dfrac{\text{remainder}}{Q(x)}$. As $x \to \pm\infty$ the remainder term vanishes, so the line is the oblique asymptote.

A horizontal-asymptote example

For $g(x) = \dfrac{2x^2 - 3}{x^2 + 4}$, the degrees are equal, so the horizontal asymptote is the ratio of leading coefficients $y = \dfrac{2}{1} = 2$. The denominator $x^2 + 4$ has no real zeros, so there are no vertical asymptotes; the $y$-intercept is $-\tfrac{3}{4}$ and the $x$-intercepts solve $2x^2 - 3 = 0$.