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Mathematics LibreTexts

Glossary

  • Page ID
    51408
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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)DefinitionImageCaptionLinkSource
    dependent variablean output variable    
    domainthe set of all possible input values for a relation    
    functiona relation in which each input value yields a unique output value    
    horizontal line testa method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once    
    independent variablean input variable    
    inputeach object or value in a domain that relates to another object or value by a relationship known as a function    
    one-to-one functiona function for which each value of the output is associated with a unique input value    
    outputeach object or value in the range that is produced when an input value is entered into a function    
    rangethe set of output values that result from the input values in a relation    
    relationa set of ordered pairs    
    vertical line testa method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once    
    even functiona function whose graph is unchanged by horizontal reflection, \(f(x)=f(−x)\), and is symmetric about the y-axis    
    horizontal compressiona transformation that compresses a function’s graph horizontally, by multiplying the input by a constant b>1    
    horizontal reflectiona transformation that reflects a function’s graph across the y-axis by multiplying the input by −1    
    horizontal shifta transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input    
    horizontal stretcha transformation that stretches a function’s graph horizontally by multiplying the input by a constant 0<b<1    
    odd functiona function whose graph is unchanged by combined horizontal and vertical reflection, \(f(x)=−f(−x)\), and is symmetric about the origin    
    vertical compressiona function transformation that compresses the function’s graph vertically by multiplying the output by a constant 0<a<1    
    vertical reflectiona transformation that reflects a function’s graph across the x-axis by multiplying the output by −1    
    vertical shifta transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output    
    vertical stretcha transformation that stretches a function’s graph vertically by multiplying the output by a constant a>1    
    absolute value equationan equation of the form \(|A|=B\), with \(B\geq0\); it will have solutions when \(A=B\) or \(A=−B\)    
    absolute value inequalitya relationship in the form \(|A|<B\), \(|A|{\leq}B\), \(|A|>B\), or \(|A|{\geq}B\)    
    decreasing linear functiona function with a negative slope: If \(f(x)=mx+b\), then \(m<0\).    
    increasing linear functiona function with a positive slope: If \(f(x)=mx+b\), then \(m>0\).    
    linear functiona function with a constant rate of change that is a polynomial of degree 1, and whose graph is a straight line    
    point-slope formthe equation for a line that represents a linear function of the form \(y−y_1=m(x−x_1)    
    slopethe ratio of the change in output values to the change in input values; a measure of the steepness of a line    
    slope-intercept formthe equation for a line that represents a linear function in the form \(f(x)=mx+b\)    
    y-interceptthe value of a function when the input value is zero; also known as initial value    
    horizontal linea line defined by \(f(x)=b\), where \(b\) is a real number. The slope of a horizontal line is 0.    
    parallel linestwo or more lines with the same slope    
    perpendicular linestwo lines that intersect at right angles and have slopes that are negative reciprocals of each other    
    vertical linea line defined by \(x=a\), where a is a real number. The slope of a vertical line is undefined.    
    x-interceptthe 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 conjugatethe 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 numberthe 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 planea 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 numbera number in the form bi where \(i=\sqrt{−1}\)    
    axis of symmetrya 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 functionthe 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 functionthe function that describes a parabola, written in the form \(f(x)=a(x−h)^2+k\), where \((h, k)\) is the vertex.    
    vertexthe point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function    
    vertex form of a quadratic functionanother name for the standard form of a quadratic function    
    zerosin a given function, the values of \(x\) at which \(y=0\), also called roots    
    coefficienta nonzero real number that is multiplied by a variable raised to an exponent (only the number factor is the coefficient)    
    continuous functiona function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph    
    degreethe highest power of the variable that occurs in a polynomial    
    end behaviorthe behavior of the graph of a function as the input decreases without bound and increases without bound    
    leading coefficientthe coefficient of the leading term    
    leading termthe term containing the highest power of the variable    
    polynomial functiona 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 functiona 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 curvea graph with no sharp corners    
    term of a polynomial functionany \(a_ix^i\) 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 pointthe location at which the graph of a function changes direction    
    global maximumhighest turning point on a graph; \(f(a)\) where \(f(a){\geq}f(x)\) for all \(x\).    
    global minimumlowest turning point on a graph; \(f(a)\) where \(f(a){\leq}f(x)\) for all \(x\).    
    Intermediate Value Theoremfor 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    
    multiplicitythe 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 Algorithmgiven 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 divisiona shortcut method that can be used to divide a polynomial by a binomial of the form \(x−k\)    
    Descartes’ Rule of Signsa 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 Algebraa polynomial function with degree greater than 0 has at least one complex zero    
    Linear Factorization Theoremallowing 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 Theoremthe 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 Theoremif a polynomial \(f(x)\) is divided by \(x−k\),then the remainder is equal to the value \(f(k)\)    
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