# Glossary

- Page ID
- 84837

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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 |
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(Eg. "Genetic, Hereditary, DNA ...") | (Eg. "Relating to genes or heredity") | The infamous double helix | https://bio.libretexts.org/ | CC-BY-SA; Delmar Larsen |

Word(s) | Definition | Image | Caption | Link | Source |
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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 \(f(k)\) |