Rational functions and graphs — common mistakes

Always draw asymptotes as dashed lines and clearly label their equations. This not only secures method marks but also provides a crucial framework for your sketch, making it much easier to draw the shape of the curve correctly.

When solving an inequality like $\frac{P(x)}{Q(x)} > k$, it's often most reliable to sketch the graphs of $y=\frac{P(x)}{Q(x)}$ and $y=k$. You can then visually identify the regions where the curve is above the line. This graphical approach avoids the algebraic pitfalls of multiplying by terms that could be negative.

A horizontal asymptote ($y=c$) occurs when the degree of the numerator is less than or equal to the degree of the denominator. An oblique asymptote ($y=mx+c$) occurs when the degree of the numerator is exactly one greater than the degree of the denominator. A graph cannot have both.

Yes, a graph can cross its horizontal or oblique asymptote. These asymptotes only describe the end behaviour of the function as x approaches positive or negative infinity. However, a graph can NEVER cross a vertical asymptote, as the function is undefined at that x-value.

For a full, detailed sketch, yes. Stationary points are crucial for showing the correct shape of the curve, such as local maxima and minima. However, if an exam question only asks for asymptotes and intercepts, you may not need to find them. Always read the question carefully to see what features are required.

If a factor $(x-a)$ cancels out, the graph has a 'hole' (a point of removable discontinuity) at $x=a$, not a vertical asymptote. The graph will look like the simplified function, but with an open circle at the point where $x=a$ to show it's undefined there.