Revision sheet

Asymptotes of rational functions

• Vertical: zeros of the denominator.
• Horizontal: compare degrees. Top smaller: $y = 0$. Equal: ratio of leading coefficients. Top larger: none.

Watch out: use the leading coefficients, not the constants.

Intercepts and holes

1. Cancel common factors: each one leaves a hole.
2. Hole height: the simplified rule at that $x$.
3. $x$-intercepts: remaining zeros of the numerator; $y$-intercept: $f(0)$.

Watch out: zeros of the denominator give asymptotes, not intercepts.

Multiplying and dividing rational expressions

Factor, cancel common factors, multiply; divide by multiplying by the reciprocal.

Watch out: cancel factors, never terms.

Adding rational expressions

Common denominator, then add the numerators.

Watch out: never add the denominators.

Rational equations

Clear denominators → solve → check in the original.

Watch out: reject values that make a denominator 0.

Radical equations

Isolate → square (or cube) → solve → check.

Watch out: squaring can add extraneous solutions.