Ch. 5 Key Terms - Algebra and Trigonometry | OpenStax

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Key Terms

arrow notation: a way to represent symbolically the local and end behavior of a function by using arrows to indicate that an input or output approaches a value

axis of symmetry: a vertical line drawn through the vertex of a parabola, that opens up or down, around which the parabola is symmetric; it is defined by $x = - \frac{b}{2 a} .$

coefficient: a nonzero real number multiplied by a variable raised to an exponent

constant of variation: the non-zero value $k$ that helps define the relationship between variables in direct or inverse variation

continuous function: a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph

degree: the highest power of the variable that occurs in a polynomial

Descartes’ Rule of Signs: a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of $f (x)$ and $f (- x)$

direct variation: the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other

Division Algorithm: given a polynomial dividend $f (x)$ and a non-zero polynomial divisor $d (x)$ where the degree of $d (x)$ is less than or equal to the degree of $f (x)$ , there exist unique polynomials $q (x)$ and $r (x)$ such that $f (x) = d (x) q (x) + r (x)$ where $q (x)$ is the quotient and $r (x)$ is the remainder. The remainder is either equal to zero or has degree strictly less than $d (x) .$

end behavior: the behavior of the graph of a function as the input decreases without bound and increases without bound

Factor Theorem: $k$ is a zero of polynomial function $f (x)$ if and only if $(x - k)$ is a factor of $f (x)$

Fundamental Theorem of Algebra: a polynomial function with degree greater than 0 has at least one complex zero

general form of a quadratic function: the function that describes a parabola, written in the form $f (x) = a x^{2} + b x + c$ , where $a, b,$ and $c$ are real numbers and $a \neq 0.$

global maximum: highest turning point on a graph; $f (a)$ where $f (a) \geq f (x)$ for all $x .$

global minimum: lowest turning point on a graph; $f (a)$ where $f (a) \leq f (x)$ for all $x .$

horizontal asymptote: a horizontal line $y = b$ where the graph approaches the line as the inputs increase or decrease without bound.

Intermediate Value Theorem: for two numbers $a$ and $b$ in the domain of $f,$ if $a < b$ and $f (a) \neq f (b),$ then the function $f$ takes on every value between $f (a)$ and $f (b)$ ; specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the $x -$ axis

inverse variation: the relationship between two variables in which the product of the variables is a constant

inversely proportional: a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases

invertible function: any function that has an inverse function

joint variation: a relationship where a variable varies directly or inversely with multiple variables

leading coefficient: the coefficient of the leading term

leading term: the term containing the highest power of the variable

Linear Factorization Theorem: allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form $(x - c)$ , where $c$ is a complex number

multiplicity: the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form ${(x - h)}^{p}$ , $x = h$ is a zero of multiplicity $p .$

polynomial function: a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

power function: a function that can be represented in the form $f (x) = k x^{p}$ where $k$ is a constant, the base is a variable, and the exponent, $p$ , is a constant

rational function: a function that can be written as the ratio of two polynomials

Rational Zero Theorem: the possible rational zeros of a polynomial function have the form $\frac{p}{q}$ where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

Remainder Theorem: if a polynomial $f (x)$ is divided by $x - k$ , then the remainder is equal to the value $f (k)$

removable discontinuity: a single point at which a function is undefined that, if filled in, would make the function continuous; it appears as a hole on the graph of a function

roots: in a given function, the values of $x$ at which $y = 0$ , also called zeros

smooth curve: a graph with no sharp corners

standard form of a quadratic function: the function that describes a parabola, written in the form $f (x) = a {(x - h)}^{2} + k$ , where $(h, k)$ is the vertex

synthetic division: a shortcut method that can be used to divide a polynomial by a binomial of the form $x - k$

term of a polynomial function: any $a_{i} x^{i}$ of a polynomial function in the form $f (x) = a_{n} x^{n} + ... + a_{2} x^{2} + a_{1} x + a_{0}$

turning point: the location at which the graph of a function changes direction

varies directly: a relationship where one quantity is a constant multiplied by the other quantity

varies inversely: a relationship where one quantity is a constant divided by the other quantity

vertex: the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function

vertex form of a quadratic function: another name for the standard form of a quadratic function

vertical asymptote: a vertical line $x = a$ where the graph tends toward positive or negative infinity as the inputs approach $a$

zeros: in a given function, the values of $x$ at which $y = 0$ , also called roots

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Authors: Jay Abramson
Publisher/website: OpenStax
Book title: Algebra and Trigonometry
Publication date: Feb 13, 2015
Location: Houston, Texas
Book URL: https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites
Section URL: https://openstax.org/books/algebra-and-trigonometry/pages/5-key-terms

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