Revision sheet

End behavior

• Only the leading term (highest power) matters for large $|x|$.
• Even degree: both ends the same way. Odd degree: opposite ways.
• A negative leading coefficient flips the picture.

Watch out: the leading term is not always written first.

Quadratic functions

1. Vertex at $x = -\dfrac{b}{2a}$.
2. Its height $f\left(-\dfrac{b}{2a}\right)$ is the minimum ($a > 0$) or maximum ($a < 0$).
3. Vertex form: $a(x - h)^2 + k$, vertex $(h, k)$.

Watch out: the constant term $c$ is $f(0)$, not the extreme value.

Zeros and multiplicity

• $(x - r)$ gives the zero $r$; $(x + q)$ gives the zero $-q$.
• Odd multiplicity: the graph crosses. Even: it touches and turns back.

Watch out: check the sign: $x + 3 = 0$ means $x = -3$.

Remainder and factor theorems

• Remainder of $p(x) \div (x - r)$: $p(r)$.
• $x - r$ is a factor $\iff p(r) = 0$.
• For $x + r$, evaluate at $-r$.

Watch out: a factor needs a remainder of $0$.

All zeros from one zero

1. Check $p(r) = 0$.
2. Divide by $x - r$: $p(x) = (x - r)\,q(x)$.
3. Solve $q(x) = 0$; list $r$ with those solutions.

Watch out: $x^2 = 9$ gives $3$ and $-3$.

Complex zeros

• $i^2 = -1$, so $\sqrt{-k^2} = ki$.
• $b^2 - 4ac < 0$: no real zeros, two complex ones.
• Real coefficients: $p + qi$ and $p - qi$ come together.

Watch out: the conjugate changes only the sign of the $i$ part.

Complex arithmetic

Add parts separately; multiply out and use $i^2 = -1$; conjugates give a real product.

Watch out: $i^2$ is $-1$, not $1$.

Complex solutions

Discriminant < 0 → $x = p \pm qi$ (conjugates).

Watch out: divide the whole numerator by $2a$.

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.

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.