AP Precalculus Unit 1 Rational Functions: Structure, Asymptotes, Holes, and Solving

Rational function

A function that can be written as a ratio of two polynomials: f(x)=P(x)/Q(x), where Q(x) is not the zero polynomial.

Numerator (P(x))

The polynomial on top of a rational function; its zeros can affect intercepts and sign, and (if shared) can cancel with denominator factors to create holes.

Denominator (Q(x))

The polynomial on the bottom of a rational function; its zeros make the function undefined and determine domain restrictions (and may create vertical asymptotes or holes).

Domain (of a rational function)

All real numbers except values that make the denominator equal to zero.

Domain restriction / excluded value

An x-value that must be excluded because it makes an original denominator zero, even if a factor later cancels in simplification.

Slant (oblique) asymptote

A line $y=mx+b$ ($m \neq 0$) that the graph approaches as $x \rightarrow \pm \infty$, typically when $\text{deg(numerator)}=\text{deg(denominator)}+1$.

Candidate vertical asymptote

A value x=a where the denominator is zero before any cancellation; it becomes a true vertical asymptote only if the factor does not cancel.

One-sided behavior

How a function behaves as $x \rightarrow \pm \infty$; for polynomials it is determined by the leading term.

Factored form

Writing polynomials as products of factors (e.g., (x-3)(x+1)), which makes it easier to see cancellations, holes, and vertical asymptotes.

Common factor (in rational functions)

A factor that appears in both numerator and denominator; canceling it changes the formula but not the original domain restrictions.

Hole

A removable discontinuity caused by a factor canceling; the graph has a missing point (open circle) rather than blowing up to infinity.

Removable discontinuity

A discontinuity where the function is undefined at a point but approaches a finite value there (a hole).

Open circle

The graphing symbol for a hole, marking a point (a, f_simplified(a)) that is not included in the original function.

End behavior

The behavior of a function as x→∞ and x→−∞; for rational functions it is largely determined by leading terms and degrees.

Leading term

The highest-degree term of a polynomial (e.g., $3x^2$ in $3x^2 - 5$); it dominates the polynomial for large |x|.

Degree (of a polynomial)

The highest exponent of x in the polynomial; comparing degrees of numerator and denominator predicts horizontal/slant asymptotes.

Horizontal asymptote

A constant line y=c that a rational function approaches as x→∞ and/or x→−∞ (often found by degree comparison).

Slant (oblique) asymptote

A line y=mx+b (m≠0) that the graph approaches as x→±∞, typically when deg(numerator)=deg(denominator)+1.

Polynomial asymptote

A non-linear asymptote (degree $> 1$) that can occur when $\text{deg(numerator)} > \text{deg(denominator)} + 1$, found via long division.

Polynomial long division (for rational functions)

A method to rewrite P(x)/Q(x)=S(x)+R(x)/Q(x), used to find slant or higher-degree asymptotes.

Quotient S(x)

The result of dividing $P(x)$ by $Q(x)$; when $\text{deg}(P)=\text{deg}(Q)+1$, $S(x)$ is the slant asymptote.

Remainder term R(x)/Q(x)

The leftover fraction after division; as $x \rightarrow \pm\infty$ it approaches $0$, so the graph approaches $y = S(x)$.

Least common denominator (LCD)

The smallest expression containing all denominator factors; multiplying by the LCD clears fractions when solving rational equations.

Extraneous solution

A value that appears after algebraic steps (often multiplying by an expression in x) but does not satisfy the original equation or violates restrictions.

Sign analysis (sign chart / test intervals)

A method for solving rational inequalities by finding critical numbers (zeros of numerator/denominator) and testing the sign on each interval.