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Mathematics LibreTexts

Glossary

  • Page ID
    84837
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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 \(f(k)\)        
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