Algebra Vocabulary List (Definitions for Middle School Teachers)


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1 Algebra Vocabulary List (Defiitios for Middle School Teachers) A Absolute Value Fuctio The absolute value of a real umber x, x is xifx 0 x = xifx < 0 Algebra Lab Gear a set of maipulatives that are desiged to represet polyomial expressios. The set icludes represetatios for positive/egative 1, 5, 25, x, 5x, y, 5y, xy, x 2, y 2, x 3, y 3, x 2 y, xy 2. The maipulatives ca be used to model additio, subtractio, multiplicatio, divisio, ad factorig of polyomials. They ca also be used to model how to solve liear equatios Algebra Tiles a set of maipulatives that are desiged for modelig algebraic expressios visually. Each tile is a geometric model of a term. The set icludes represetatios for positive/egative 1, x, ad x 2. The maipulatives ca be used to model additio, subtractio, multiplicatio, divisio, ad factorig of polyomials. They ca also be used to model how to solve liear equatios. Algebraic Expressio a umber, variable, or combiatio of the two coected by some mathematical operatio like additio, subtractio, multiplicatio, divisio, expoets ad/or roots. _simp.htm Area Uder the Curve suppose the curve y=f(x) lies above the x axis for all x i [a, b]. The area uder the curve is the area of the regio bouded by the curve, the x axis ad the lies x=a ad x=b. It equals f ( xdx ). b a
2 Arithmetic Sequece (arithmetic progressio) A sequece of umbers i which the differece of two cosecutive terms is the same. A sequece with a geeral term a = 1 a + + d or a = a1 + ( 1) d is called a arithmetic sequece. Example: let d=2 ad a 1 = 1, the 1,3,5,7, forms a arithmetic sequece with first term equal to 1 ad commo differece(differece betwee ay two cosecutive terms) equal to 2. o For more Ifo: Arithmetic Series the idicated sum of the terms of a arithmetic sequece. Example of fiite arithmetic series: (Note this is the sum of a fiite arithmetic sequece whose first term is 1 ad commo differece is 2). I geeral, the sum of a fiite arithmetic series is i = 1 a = ( a + a ) / 2 i 1 o For more Ifo: B Biomial a expressio cosistig of two terms, such as 2x + 5y or 7x C Coefficiet the umerical part of a term, usually writte before the literal part, as 2 i 2x or 2(x + y). Most commoly used i algebra for the costat factors, as distiguished from the variables. Coefficiet Matrix (matrix of the coefficiets) the matrix of coefficiets of the variables i a system of equatios writte i stadard form. For example: the coefficiet matrix for 2x 3y = 8 ad 4x + 5y = 2 is deoted by
3 Compressed Graph (vertical shrik) a shrik i which a plae figure is distorted vertically. It s a trasformatio i which all distaces o the coordiate plae are shorteed by multiplyig all y coordiates of a graph by a commo factor less tha 1. Costat a symbol represetig a value that does t chage. Cotiuous Curve ca be depicted as, path of a cotiuously movig poit. A lie or curve that exteds without a break or abrupt chages. Cramer s Rules a rule usig determiats to express the solutio of a system of liear algebraic equatios for which the umber of equatios is equal to the umber of variables. e f e f bd ac For example: To solve x ad y from ax + by = e; cx + dy = f ; x= ; y =, ab ab cd cd where ab ad bc cd =. Cubic Fuctio a polyomial fuctio of degree 3, usually writte i the form y = ax 3 + bx 2 + cx + d, where a, b, c ad d are costats. MGraphFCubic.html uctios.html D Degree of a Moomial the degree of a term i oe variable is the expoet of that variable; the degree of a term i several variables is equal to the sum of the expoets of its variables. For example, the degree of 4x 2 is 2 ad the degree of 4x 2 y 3 z 2 is 7.
4 Degree of a Polyomial the degree of the moomial of largest degree belogig to that polyomial. For example, the degree of a polyomial, 4x 5 + 6x is 5. Determiat is a special set of mathematical operatios associated with a square array. The result of the operatio is a scalar value. The determiat below has two rows ad two colums ad is called a secodorder determiat. a b c d A secodorder determiat is evaluated as follows. a b Value of a SecodOrder Determiat: = ad bc c d Notice that the value of the determiat is foud by calculatig the differece of the products of the two diagoals. a b bc ad  bc c d ad Differece of Squares a differece of two squares ca be represeted i a expressio of the form a 2 b 2 ad factors ito the form (a + b)(a b). o For more Ifo: Direct Variatio a liear fuctio of the form y = cx, where c is the costat of variatio; c 0. We say that y is directly proportioal to x, i.e. y varies directly as x varies. If x is doubled, tripled or halved, the y is also doubled, tripled, or halved. If x icreases oe uit, the y icreases c uits. Domai of a Fuctio The set of all possible iput values of a fuctio. Give a real 1 valued fuctio f( x) =, its domai is the set of all real umbers excludig 0. x
5 E Equatio A statemet assertig the equality of two expressios that are separated ito left ad right sides ad joied by a equal sig. Expoetial Fuctio a equatio of the form f(x) = a b x + k where a 0, b > 0, b 1 ad x is ay real umber, is called a expoetial fuctio with base b. F FOIL method a applicatio of the distributive property used to multiply two biomials. The product of the two biomials is foud by multiplyig the First, Outer, Ier, ad Last terms. Fuctio a set of ordered pairs such that o two ordered pairs have the same first member. A relatio, such that each elemet of a set (the domai) is associated with a uique elemet of aother (possibly the same) set (the codomai ot to be cofused with the rage). Fuctio Notatio fuctio otatio uses f(x) (or g(x), h(x), etc.), istead of y, to represet the depedet variable. G Geometric Sequece (geometric progressio) is a sequece of umbers i which each term is obtaied by multiplyig the precedig term by the same umber (commo ratio). The followig is a geometric progressio: 1, 2, 4, 8, 16, 32 The commo ratio for this geometric progressio is Geometric Series The idicated sum of the terms of a geometric sequece. The geometric series correspodig to the geometric sequece: 1, 2, 4, 8, 16, 32 is
6 Graphs a pictorial represetatio of some mathematical relatioship. It ca be a poit o a umber lie, which is a graph of a real umber. H Horizotal Asymptote if there exists a umber c such that the curve f(x) approaches c as x approaches or f(x) approaches c as x approaches, the the lie y=c is called a horizotal asymptote of the curve f(x). I Idetity Fuctio is a fuctio i the form f(x) = x. More geerally, a idetity fuctio is oe which does ot chage the domai values at all. Iequality A iequality is a statemet with the symbol <,, >, or betwee umerical or variable expressios. Ifiite Series A ifiite series is a ifiite, ordered set of terms combied together by the additio operator. For example: Iterval o a Real Number Lie is the set cotaiig all umbers betwee two give umbers (the ed poits of the iterval) icludig oe, both, or either ed poit. html Iverse Fuctio if f is a oe to oe ad oto fuctio from a set X to a set Y, the the correspodece that goes back from Y to X is also a fuctio ad is kow as the iverse fuctio. For the defiitio of oto ad oe to oe fuctios please look uder O.
7 Iverse Variatio A variatio stated i the form y = k, where k is the costat of x variatio. I this equatio the values of y get smaller as the vales of x get greater. It is said that x ad y are iversely proportioal. Iteratio repetitio of a sequece of istructios. Iteratio is characterized by a set of iitial coditios, a iterative step ad a termiatio coditio. For example, the set of Fiboacci umbers is geerated by begiig with 1, 1, ad usig the iterative process of summig the two previous terms to determie all subsequet terms. The set of Fiboacci umbers is {1, 1, 2, 3, 5, 8, 13...} L Limit (Limit of a fuctio i oe variable) a limit is the value of the fuctio as the variable approaches a particular poit. For example the limit of the fuctio x as x approaches 4 is 2. The limit of 1/ as approaches ifiity is 0. o Liear Fuctio a fuctio whose graph is a straight lie. A liear fuctio of x ca be writte i the form f(x) = mx + b, where m ad b are costats. Logarithmic Fuctio The logarithmic fuctio f(x) = log b x is the iverse of the expoetial fuctio f(x) = b x, where b > 0 ad b 1. A fuctio defied by a expressio of the form log f(x). M Matrix a rectagular array of variables or costats i horizotal rows ad vertical colums, usually eclosed i brackets. Moomial a expressio that ca be a costat, a variable, or a product of a costat ad oe or more variable. Each of the followig is a moomial. 5 (a costat) 3z (a product of a costat ad oe variable) x (a variable) 6xyz (a product of a costat ad more tha oe variable)
8 N Natural Expoet (e) e e is the limit of 1 + as approaches ifiity. (1 ) NoLiear Fuctio  A fuctio whose graph is ot a straight lie. Example: Polyomials of degree two or higher, expoetial fuctio, sie, cosie fuctios etc. ml O OetoOe Fuctio a fuctio f is said to be oe to oe (ijective) if ad oly if f(x) = f(y) implies x = y. i.e. o two distict elemets i the domai correspod to the same elemet i the Rage. P Parabola The geeral shape of the graph of a quadratic fuctio. The set of all poits i a plae that are the same distace from a give poit called the focus ad a give lie called the directrix. Paret Fuctio the basic fuctio for a family of fuctios. For istace, y = x 2 is the paret fuctio of the family of quadratic fuctios. Polyomial Sum ad/or differece of terms: example: 3x 2 ad 4x 25x + 3 A polyomial with two terms is a biomial. A polyomial with three terms is a triomial Polyomial Fuctio a fuctio of the form Px ( ) = ax + a x +... ax+ a for all real x, where the coefficiets are real umbers ad a o egative iteger. If a 0, P(x) is called a real polyomial of degree. itio.html
9 Proportio the statemet of equality of two ratios; a equatio statig that two ratios are equal. Q Quadratic Fuctio a polyomial fuctio of degree 2, ad i the form f(x) = ax 2 + bx + c. tml R Rage (rage of a fuctio) The set of possible values which a fuctio's output ca be. Give the fuctio {(1, 5), (2, 10), (13, 15), (4, 20)} its rage is {5, 10, 15, 20}. Rate of Chage the ratio of the chage i oe quatity to a correspodig uit chage i aother quatity. p( x) Ratioal Fuctio divisio of two polyomial fuctios i the form f( x) =, where qx ( ) p(x) ad q(x) are polyomial fuctios ad q(x) 0. (The domai of f cosists of all real umbers x such that the deomiator q(x) is ot equal to 0.) tml Recursive Formula a recursive formula has two parts: the value(s) of the first term(s), ad a recursio equatio that shows how to fid each term from the term(s) before it. Recursive Sequece a recursive sequece is a ordered list of umbers defied by a startig value ad a rule, applyig the rule agai to the previous value, ad repeatig this process.
10 Relatio A relatio is ay subset of a Cartesia product. For istace, a subset of A B, called a "biary relatio from A to B," is a collectio of ordered pairs (a, b) with first compoets from A ad secod compoets from B, ad, i particular, a subset of A A is called a "relatio o A." Root a solutio of a equatio. Roots of the Equatio a umber which, whe substituted for the variable i the equatio satisfies the equatio. For example: 2 is a root for the equatio x 2 4= 0, but 3 is ot. S Sequece of Numbers A ordered arragemet of umbers. We deote a sequece as a where its th term is a = 1. For example: cosider the sequece 1,,,... it ca also be expressed as 1. = 1 Slope the ratio of rise to ru for a lie i the coordiate plae. The slope of a lie described by f(x) = mx + b is m. Slope of the Taget Lie the slope of the lie that is taget to the fuctio graph at a certai poit (c, f(c)). (The lie that passes through the poit (c, f(c)) with slope f ' () c is called the taget lie at the poit (c, f(c)). Stretched Graph A trasformatio i which all distaces o the coordiate plae are legtheed by multiplyig either all xcoordiates (horizotal dilatio) or all ycoordiates (vertical dilatio) of a graph by a commo factor greater tha 1. Symmetry about the Xaxis if for ay poit (x, y) o a graph, the poit (x, y) is also o the graph, the the graph is said to be symmetric with respect to the Xaxis. Also, if the equatio of the curve is ualtered whe y is replaced by y. i.e. f ( xy, ) = f( x, y).
11 Symmetry about the Yaxis if for ay poit (x, y) o a graph, the poit (x, y) is also o the graph, the the graph is said to be symmetric with respect to the Yaxis. Also, if the equatio of the curve is ualtered whe x is replaced by x.i.e. f ( xy, ) = f( xy, ) System of Equatios two or more equatios i two or more variables cosidered together or simultaeously. The equatios i the system may or may ot have a commo solutio. T Table of Values (Table) a systematic listig of results already worked out. Traslatio a trasformatio i which a figure is moved from oe locatio to aother o the coordiate plae without chagig its size, shape, or orietatio. Triomial a polyomial with three ulike terms. For example, x 2 3x TwoColor Couters a diskshaped maipulative that is geerally white o oe side ad red o the other side. These couters are ofte used for itroducig iteger cocepts. The white side represets a positive oe value, ad the red side represets a egative oe value. V Variable A variable is a symbol o whose value a fuctio, polyomial, etc., depeds. For example, the variables i the fuctio f(x,y) are x ad y. A fuctio havig a sigle variable is said to be uivariate, oe havig two variables is said to be bivariate, ad oe havig two or more variables is said to be multivariate. Vertical Asymptote The lie x=c is a vertical asymptote for a fuctio f, if ay of the followig coditio holds: 1. f approaches or as x approaches c. 2. f approaches or as x approaches c from the left. 3. f approaches or as x approaches c from the right.
12 X Xitercept the xcoordiate of the poit at which a graph crosses the xaxis. Y Yitercept  the ycoordiate of the poit at which a graph crosses the yaxis. ml Z Zero (of a fuctio) for ay fuctio f(x), if f(a) = 0, the a is a zero of the fuctio. The value(s) of x for which the value of the fuctio is zero. The real zeros of a fuctio are the xitercepts of its graph i the coordiate plae.
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