Quick questions on Rational functions and graphing: VCE Specialist Mathematics Unit 3

The function is undefined where $Q(x) = 0$. If $P$ and $Q$ have no common factor there, the line $x = a$ is a vertical asymptote and the graph shoots to $\pm\infty$ on each side. If $P$ and $Q$ share the factor $(x - a)$, there is instead a hole (point discontinuity) at $x = a$.

The $y$-intercept is $f(0)$ (when $0$ is in the domain). The $x$-intercepts are the zeros of the numerator that are not also zeros of the denominator, that is, the solutions of $P(x) = 0$ with $Q(x) \neq 0$.

State all asymptotes of $f(x) = \dfrac{3x - 6}{x + 1}$. [2 marks]

Find the oblique asymptote of $f(x) = \dfrac{2x^2 + x}{x - 1}$. [2 marks]

Describe how $y = |x^2 - 1|$ differs from $y = x^2 - 1$. [2 marks]