Topic 1.11: Equivalent Representations of Polynomial and Rational Functions

Vocabulary practice flashcards based on Topic 1.11 covering characteristics of rational functions, asymptotes, polynomial division, Pascal's Triangle, and the Binomial Theorem.

Hole (Removable Discontinuity)

A point of discontinuity in a rational function $h(x) = \frac{f(x)}{g(x)}$ occurring at an $x$-value that is a common zero of both the numerator and the denominator.

Vertical Asymptote

A vertical line $x = c$ approached by the graph of a rational function, occurring where there is a zero in the denominator that is not a zero in the numerator (or where the multiplicity of the zero in the denominator exceeds that in the numerator).

Horizontal Asymptote (Bottom-Heavy)

The horizontal line $y = 0$, which the graph of a rational function approaches as $x \rightarrow \pm\infty$ when the degree of the denominator is strictly greater than the degree of the numerator.

Horizontal Asymptote (Equal Degrees)

The horizontal line $y = \frac{a}{b}$, which a rational function approaches as $x \rightarrow \pm\infty$ when the degrees of the numerator and denominator are equal, given by the ratio of their leading coefficients.

Slant Asymptote

A linear, non-horizontal line of the form $y = mx + b$ ($m \neq 0$) that a rational function approaches as $x \rightarrow \pm\infty$, occurring when the degree of the numerator polynomial is exactly one greater than the degree of the denominator polynomial.

Polynomial Long Division

An algebraic method used to divide a polynomial $f(x)$ by a polynomial $g(x)$ to yield a quotient $q(x)$ and a remainder $r(x)$, written as $\frac{f(x)}{g(x)} = q(x) + \frac{r(x)}{g(x)}$, where the degree of $r(x)$ is strictly less than the degree of $g(x)$.

Zero of a Rational Function

An input value $x$ for which the output of the rational function is 0, occurring at an $x$-value that makes the numerator equal to zero while the denominator is non-zero.

Domain of a Rational Function

The set of all real numbers except those values of $x$ that cause the denominator polynomial to equal zero (the locations of holes and vertical asymptotes).

Al-Karaji and Jia Xian

The mathematicians who separately developed the first formulations of the binomial theorem around the year $1000\,\text{AD}$.

Pascal's Triangle

A triangular array of numbers whose rows provide the coefficients for the expansion of a binomial, where the value of $\binom{n}{r}$ corresponds to the $r^{\text{th}}$ element of the $n^{\text{th}}$ row (starting with Row 0 and the $0^{\text{th}}$ element).

The Binomial Theorem

The algebraic theorem stating that for any positive integer $n$, $(a+b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b^1 + \binom{n}{2}a^{n-2}b^2 + \dots + \binom{n}{n-1}a^1b^{n-1} + \binom{n}{n}b^n$.

Degree Pattern in Binomial Expansion

The structural pattern in expanding $(a+b)^n$ where the exponent of $a$ starts at $n$ and decreases by 1 in each successive term, while the exponent of $b$ starts at 0 and increases by 1 in each term until reaching $n$.

Combination

A mathematical operation denoted by $\binom{n}{r}$, $_nC_r$, or $C(n, r)$, representing the number of ways to choose $r$ items from a set of $n$ items, defined algebraically as $\frac{n!}{r!(n-r)!}$.