Glossary

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Apr 2, 2022
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Example and Directions
Words (or words that have the same definition)	The definition is case sensitive	(Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pages]	(Optional) Caption for Image	(Optional) External or Internal Link	(Optional) Source for Definition
(Eg. "Genetic, Hereditary, DNA ...")	(Eg. "Relating to genes or heredity")		The infamous double helix	https://bio.libretexts.org/	CC-BY-SA; Delmar Larsen

Glossary Entries
Word(s)	Definition	Image	Caption	Link	Source
dependent variable	an output variable
domain	the set of all possible input values for a relation
function	a relation in which each input value yields a unique output value
horizontal line test	a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once
independent variable	an input variable
input	each object or value in a domain that relates to another object or value by a relationship known as a function
one-to-one function	a function for which each value of the output is associated with a unique input value
output	each object or value in the range that is produced when an input value is entered into a function
range	the set of output values that result from the input values in a relation
relation	a set of ordered pairs
vertical line test	a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once
even function	a function whose graph is unchanged by horizontal reflection, $f(x)=f(−x)$ , and is symmetric about the y-axis
horizontal compression	a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant b>1
horizontal reflection	a transformation that reflects a function’s graph across the y-axis by multiplying the input by −1
horizontal shift	a transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input
horizontal stretch	a transformation that stretches a function’s graph horizontally by multiplying the input by a constant 0<b<1
odd function	a function whose graph is unchanged by combined horizontal and vertical reflection, $f(x)=−f(−x)$ , and is symmetric about the origin
vertical compression	a function transformation that compresses the function’s graph vertically by multiplying the output by a constant 0<a<1
vertical reflection	a transformation that reflects a function’s graph across the x-axis by multiplying the output by −1
vertical shift	a transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output
vertical stretch	a transformation that stretches a function’s graph vertically by multiplying the output by a constant a>1
absolute value equation	an equation of the form $\|A\|=B$ , with $B\geq0$ ; it will have solutions when $A=B$ or $A=−B$
absolute value inequality	a relationship in the form $\|A\|<B$ , $\|A\|{\leq}B$ , $\|A\|>B$ , or $\|A\|{\geq}B$
decreasing linear function	a function with a negative slope: If $f(x)=mx+b$ , then $m<0$ .
increasing linear function	a function with a positive slope: If $f(x)=mx+b$ , then $m>0$ .
linear function	a function with a constant rate of change that is a polynomial of degree 1, and whose graph is a straight line
point-slope form	the equation for a line that represents a linear function of the form \(y−y_1=m(x−x_1)
slope	the ratio of the change in output values to the change in input values; a measure of the steepness of a line
slope-intercept form	the equation for a line that represents a linear function in the form $f(x)=mx+b$
y-intercept	the value of a function when the input value is zero; also known as initial value
horizontal line	a line defined by $f(x)=b$ , where $b$ is a real number. The slope of a horizontal line is 0.
parallel lines	two or more lines with the same slope
perpendicular lines	two lines that intersect at right angles and have slopes that are negative reciprocals of each other
vertical line	a line defined by $x=a$ , where a is a real number. The slope of a vertical line is undefined.
x-intercept	the point on the graph of a linear function when the output value is 0; the point at which the graph crosses the horizontal axis
complex conjugate	the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number
complex number	the sum of a real number and an imaginary number, written in the standard form $a+bi$ , where $a$ is the real part, and $bi$ is the imaginary part
complex plane	a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number
imaginary number	a number in the form bi where $i=\sqrt{−1}$
axis of symmetry	a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by $x=−\frac{b}{2a}$ .
general form of a quadratic function	the function that describes a parabola, written in the form $f(x)=ax^2+bx+c$ , where $a,b,$ and $c$ are real numbers and a≠0.
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.
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
zeros	in a given function, the values of $x$ at which $y=0$ , also called roots
coefficient	a nonzero real number that is multiplied by a variable raised to an exponent (only the number factor is the coefficient)
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
end behavior	the behavior of the graph of a function as the input decreases without bound and increases without bound
leading coefficient	the coefficient of the leading term
leading term	the term containing the highest power of the variable
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 $f(x)=kx^p$ where $k$ is a constant, the base is a variable, and the exponent, $p$ , is a constant
smooth curve	a graph with no sharp corners
term of a polynomial function	any $a_ix^i$ of a polynomial function in the form $f(x)=a_nx^n+a_{n-1}x^{n-1}...+a_2x^2+a_1x+a_0$
turning point	the location at which the graph of a function changes direction
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 $x$ .
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 functionf 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
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 $p$ .
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 $d(x)$ .
synthetic division	a shortcut method that can be used to divide a polynomial by a binomial of the form $x−k$
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 $f(−x)$
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
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
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 $f(k)$

absolute value equation | an equation of the form $|A|=B$ , with $B\geq0$ ; it will have solutions when $A=B$ or $A=−B$

absolute value inequality | a relationship in the form $|A|<B$ , $|A|{\leq}B$ , $|A|>B$ , or $|A|{\geq}B$

axis of symmetry | a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by $x=−\frac{b}{2a}$ .

coefficient | a nonzero real number that is multiplied by a variable raised to an exponent (only the number factor is the coefficient)

complex conjugate | the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number

complex number | the sum of a real number and an imaginary number, written in the standard form $a+bi$ , where $a$ is the real part, and $bi$ is the imaginary part

complex plane | a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number

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

decreasing linear function | a function with a negative slope: If $f(x)=mx+b$ , then $m<0$ .

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

dependent variable | an output variable

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 $f(−x)$

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 $d(x)$ .

domain | the set of all possible input values for a relation

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

even function | a function whose graph is unchanged by horizontal reflection, $f(x)=f(−x)$ , and is symmetric about the y-axis

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

function | a relation in which each input value yields a unique output value

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 $f(x)=ax^2+bx+c$ , where $a,b,$ and $c$ are real numbers and a≠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 compression | a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant b>1

horizontal line | a line defined by $f(x)=b$ , where $b$ is a real number. The slope of a horizontal line is 0.

horizontal line test | a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once

horizontal reflection | a transformation that reflects a function’s graph across the y-axis by multiplying the input by −1

horizontal shift | a transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input

horizontal stretch | a transformation that stretches a function’s graph horizontally by multiplying the input by a constant 0<b<1

imaginary number | a number in the form bi where $i=\sqrt{−1}$

increasing linear function | a function with a positive slope: If $f(x)=mx+b$ , then $m>0$ .

independent variable | an input variable

input | each object or value in a domain that relates to another object or value by a relationship known as a function

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 functionf 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

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

linear function | a function with a constant rate of change that is a polynomial of degree 1, and whose graph is a straight line

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 $p$ .

odd function | a function whose graph is unchanged by combined horizontal and vertical reflection, $f(x)=−f(−x)$ , and is symmetric about the origin

one-to-one function | a function for which each value of the output is associated with a unique input value

output | each object or value in the range that is produced when an input value is entered into a function

parallel lines | two or more lines with the same slope

perpendicular lines | two lines that intersect at right angles and have slopes that are negative reciprocals of each other

point-slope form | the equation for a line that represents a linear function of the form \(y−y_1=m(x−x_1)

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 $f(x)=kx^p$ where $k$ is a constant, the base is a variable, and the exponent, $p$ , is a constant

range | the set of output values that result from the input values in a relation

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.

relation | a set of ordered pairs

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

slope | the ratio of the change in output values to the change in input values; a measure of the steepness of a line

slope-intercept form | the equation for a line that represents a linear function in the form $f(x)=mx+b$

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 $a_ix^i$ of a polynomial function in the form $f(x)=a_nx^n+a_{n-1}x^{n-1}...+a_2x^2+a_1x+a_0$

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

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 compression | a function transformation that compresses the function’s graph vertically by multiplying the output by a constant 0<a<1

vertical line | a line defined by $x=a$ , where a is a real number. The slope of a vertical line is undefined.

vertical line test | a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once

vertical reflection | a transformation that reflects a function’s graph across the x-axis by multiplying the output by −1

vertical shift | a transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output

vertical stretch | a transformation that stretches a function’s graph vertically by multiplying the output by a constant a>1

x-intercept | the point on the graph of a linear function when the output value is 0; the point at which the graph crosses the horizontal axis

y-intercept | the value of a function when the input value is zero; also known as initial value

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

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