a + bi form The form of a complex number where a and b are real numbers, and i = 1.

Transcription

1 NYS Mathematics Glossary* lgebra /Trig *This glossary has bee ameded from the full SED Commecemet Level Glossary of Mathematical Terms (available at to list oly terms idicated to be at the lgebra /Trig level.) This Glossary, iteded for teacher use oly, provides a uderstadig of the mathematical terms used i the Regets-approved course etitled lgebra /Trig (as reflected i the NYS Mathematics Core Curriculum). a + bi form The form of a complex umber where a ad b are real umbers, ad i =. abscissa The horizotal or x-coordiate of a two-dimesioal coordiate system. absolute value The distace from 0 to a umber o a umber lie. The absolute value of a umber is idicated by. =, + =, ad 0=0. absolute value equatio equatio cotaiig the absolute value of a variable. x + = 9 absolute value fuctio fuctio cotaiig the absolute fuctio of a variable. x, x 0 f( x) = x = x, x< 0 absolute value iequality iequality cotaiig the absolute value of a variable. x + < 9 adjacet agles Two coplaar agles that share a commo vertex ad a commo side but have o commo iterior poits. I the figure, O ad OC are a pair of adjacet agles, but OC ad OD are ot adjacet. C O D

2 adjacet sides Two sides of ay polygo that share a commo vertex. algebraic equatio mathematical statemet that is writte usig oe or more variables ad costats which cotais a equal sig. Examples: y + = x = log ( x ) = x = 8 algebraic expressio mathematical phrase that is writte usig oe or more variables ad costats, but which does ot cotai a relatio symbol ( <, >,,, =, ) Examples: algebraic represetatio mathematical relatioship. The use of a equatio or algebraic expressio to model a algorithm a defied series of steps for carryig out a computatio or process. ambiguous case The case where the umber of triagles foud ca vary from zero to two, whe give two sides of a triagle ad the measure of the agle opposite oe of the sides. amplitude The magitude of the oscillatio of a siusoidal fuctio; the absolute value of oehalf of the differece betwee the maximum ad miimum fuctio values of a siusoidal fuctio. aalyze to examie methodically by separatig ito parts ad studyig their relatioships. agle geometric figure formed by two rays that have a commo edpoit. C C

3 agle i stadard positio agle with a vertex at the origi; the iitial ray is o the positive x-axis, ad the termial side falls i oe of the four quadrats or o oe of the axes. agle of depressio The agle formed by the horizotal ad the lie of sight whe lookig dowward. agle of depressio horizotal lie of sight object agle of elevatio The agle formed by the horizotal ad the lie of sight whe lookig upward. object lie of sight agle of elevatio horizotal atilogarithm The iverse fuctio of a logarithm; to fid a umber give its logarithm. If log 0 00 = the, is the atilogarithm of 00. approximate value value for some quatity, accurate to a specified degree. board that measures feet iches has a approximate legth to the earest foot of feet. arc legth The distace o the circumferece of a circle from oe edpoit of a arc to the other edpoit, measured alog the arc.

4 arccosie The iverse of the cosie fuctio, deoted by Note: cos x cos x cos x or arccos x. arcsie The iverse of the sie fuctio, deoted by Note: si x si x si x or arcsi x. arctaget The iverse of the taget fuctio, deoted by arc x or Note: ta x ta x arcta x. argumet The commuicatio, i verbal or writte form, of the reasoig process that leads to a valid coclusio. arithmetic sequece set of umbers i which the commo differece betwee each term ad the precedig term is costat. I the arithmetic sequece,, 8,,, the commo differece betwee each term ad the precedig term is. arithmetic series The idicated sum of a arithmetic sequece is the idicated sum of the positive odd itegers. asymptote straight lie or curve that is the limitig value of a curve.

5 base of a logarithmic fuctio The umber b i the logarithmic fuctio log b x = y where y b > 0 ad b if ad oly if b = x. base of a expoetial fuctio The umber b i the expoetial fuctio where a 0, b> 0, ad b. y x = ab, erouli experimets Probability experimets that ca be described i terms of just two outcomes; a experimet that meet the followig coditios: the experimet cosists of trials whose outcomes are either successes or failures, ad the trials are idetical ad idepedet with a costat probability of success, p, ad a costat probability of failure, q = - p. biased sample sample havig a distributio that is determied ot oly by the populatio from which it is draw, but also by some property that iflueces the distributio of the sample. poll to determie whether a stop sig is eeded at a school crossig might be biased if the sample polled cosisted oly of parets who dropped their childre off at school. bimodal data set that has two modes. biomial expasio The expasio of some power of a biomial expressio. x + y = x+ xy+ 0xy + 0xy+ xy + y. The expasio of ( ) biomial probability formula formula for determiig the probability of a eroulli experimet. biomial theorem method for expadig a biomial expressio raised to some power. k a+ b = a + a b+ a b ab + b = a b 0 k ( ) ( ) k= 0 k C0 C C... C C Ck k= 0 a+ b = a + a b+ a b + + ab + b = a b bivariate data Data ivolvig two variables. k k C ceter-radius equatio of a circle The form of the equatio of a circle with ceter (h, k) ad x h + y k =r. radius r give by the formula ( ) ( ) If the coordiates of the ceter of the circle are (, ) ad the legth of the radius is, the the equatio of the circle is ( x ) + ( y + ) =.

6 6 cetral agle agle i a circle with vertex at the ceter of the circle ad sides that are radii. Cetral agle O O circle The set of all poits (or locus of poits) i a plae that are a fixed distace, (called the radius) from a fixed poit, (called the ceter). circular fuctio fuctio that relates the coordiates of a poit o a circle to the distace from the poit to the origi. These fuctios are called circular fuctios because the coordiates of a poit o a circle are related to the lie values of trigoometric fuctios. _ r θ P (x,y) cosθ = x r y, siθ =, r = x + y r coefficiet The umerical factor of a term i a polyomial. is the coefficiet i the term x y.

7 7 cofuctio The trigoometric fuctio based o the complemet of a agle. The value of a trigoometric fuctio of a agle equals the value of the cofuctio of the complemet of the agle. Cofuctios, radias Cofuctios, degrees π si x = cos x π cos x = si x si (90 x) = cos x cos (90 x) = si x π ta x = cot x π cot x = ta x ta (90 x) = cot x cot (90 x) = ta x π sec x = csc x π csc x = sec x sec (90 x) = csc x csc (90 x) = sec x combiatio objects. arragemet of objects i which order i ot importat; a collectio of commo differece The differece betwee ay two successive terms of a arithmetic sequece commo factor umber, polyomial, or quatity that divides two or more umbers or algebraic expressios evely.,,,, are commo factors of ad 0 x is a commo factor of xy ad 6x x is a commo factor of x x ad x 6x + 8 commo logarithm logarithm to base 0. The commo logarithm of x is writte log x. For example, log 00 is sice 0 = 00. commo ratio The ratio of ay two successive terms of a geometric sequece. completig the square process used to chage a expressio of the form ax +bx +c ito a perfect square biomial by addig a suitable costat. complex fractio fractio with aother fractio i its umerator, deomiator or both. x x or x x + x

8 complex umber y umber that ca be expressed i the form a+ bi, where a ad b are real umbers ad i is the imagiary uit. 8 compositio of fuctios way of combiig fuctios i which the output of oe fuctio is used as the iput of aother fuctio; the formatio of a ew fuctio h from fuctios f ad g usig the rule hx ( ) = g f( x) = g[ f( x) ] for all x i the domai of f for which f ( x ) is i the domai of g. compoud evet evet that is derived from two or more simple evets. If we roll two dice, the the evet "gettig a six o either the first or secod die" is a compoud evet. compoud iterest method of calculatig iterest i which iterest eared is added to the pricipal ad thereafter also ears iterest. cojecture educated guess; a uprove hypothesis based o observatio, experimetatio, data collectio, etc. cojugate Two biomials whose first terms are equal ad last terms are opposites. a+ bad a b, + ad, + i ad -i costat fuctio fuctio where each elemet of the domai is mapped to the same elemet i the rage. The graph of a costat fuctio is a horizotal lie. f( x ) = cotrolled experimet experimet which compares the results obtaied from a experimetal sample agaist a cotrol sample. correlatio coefficiet umber, r, betwee - ad that idicates the stregth ad directio of the liear relatioship betwee two sets of umbers. If r=, the the poits lie o a lie which has a positive slope ad the two sets of umbers are said to be i perfect positive correlatio. If r=-, the the poits lie o a lie which has a egative slope ad the two data sets are said to be i perfect egative correlatio. cosecat For a give acute agleθ i a right triagle, the ratio of the legth of the hypoteuse of the triagle to the side opposite the acute agle. lso the reciprocal of the sie ratio of the give agle, writte as csc. See also circular fuctio. I this right triagle, csc = ad csc = C

9 cosie For a give acute agle θ i a right triagle, the ratio of the legth of the side adjacet to a acute agle to the legth of the hypoteuse. The cosie of a agle is writte as COS. See also circular fuctio. 9 I this right triagle, cos = ad cos = C cotaget For a give acute agle θ i a right triagle, the ratio of the side adjacet a acute agle to the side opposite the acute agle. The cotaget is the reciprocal of the taget ratio of the give agle, writte as cotθ. See also circular fuctio. I this right triagle, cot = ad cot = C cotermial agles gles i stadard positios that share the same termial side; formed by differet rotatios that have the same iitial ad termial sides. I each figure below, θ ad β are cotermial agles. β θ θ β The measure of θ = ad the measure of β = 9 The measure of θ = ad the measure of β =

10 0 couterexample example that disproves a geeral statemet. The statemet that the sum of two umbers is less tha their product could be refuted by the couterexample that + >. D degree measure uit of agle measure equal to 60 of a complete revolutio. degree of a moomial The sum of the expoets of the variables i the moomial. The degree of the moomial x is three. The degree of the moomial x y is five. The degree of xy is two. The degree of 7 is zero. degree of a polyomial The highest degree of ay moomial term i the polyomial. differece of two perfect squares (a b)(a + b). biomial of the form a b which ca be factored ito direct variatio relatioship i which the ratio of two variables is costat. direct variatio has a equatio i the form y = kx, where x ad y are variables ad k is the costat of variatio. discrimiat The expressio b ac where a, b ad c are the coefficiets of the quadratic equatio ax + bx + c = 0. The discrimiat is used to determie the ature of the roots of the quadratic equatio. domai The set of values of the idepedet variable for which a give fuctio is defied; the set of first coordiates i the ordered pairs of a relatio. double ad half agle formulas for trigoometric fuctios Formulas used to determie trigoometric values for double or half of a give agle. si = si cos cos = cos si = cos cos cos = si ta ta = ta cos si =± + cos cos =± cos ta =± + cos

11 double root root of a equatio that occurs twice. value r is a double root of a equatio f( x ) = 0if ( x r) is a factor of f ( x ). E e The base of the atural logarithm; a umber commoly ecoutered whe workig with expoetial fuctios to model growth, decay, cotiuously compouded iterest; e equatio mathematical setece statig that two expressios are equal. equivalet forms Differet ways of writig umbers or expressios that have equal values. 8 is equivalet to is equivalet to exact aswer The solutio to a equatio that has ot bee rouded. Whe solvig the equatio x =, the exact aswer is x =±. aswer such as.70, although correct to five decimal places, has bee rouded ad thus is ot a exact aswer. exact value The value of a expressio that has ot bee rouded. π The exact value of cos is π cos., while is a approximatio of expad a biomial power. The process of creatig a polyomial by raisig a biomial to a itegral x + y = x + x y+ xy + y. Whe expaded ( ) experimetal probability probability calculated by performig a experimet, rather tha by aalyzig a situatio mathematically; the ratio of the umber of times the evet occurs to the total umber of trials or times the activity is performed. explicit formula For a sequece a, a formula that is used to geerate the th, a, a,... a... term of a sequece.

12 expoetial form expressio or equatio cotaiig expoets. Examples: The equatio 9is the expoetial form of the logarithmic equatio log 9 =. I expoetial form, =. = expoetial fuctio fuctio with a variable i the expoet; a equatio i the form x y = ab, where a 0 ad b> 0, b. = (.0) t is a expoetial fuctio extraeous root/value equatio. solutio of a derived equatio that is ot a solutio of the origial extrapolate The process of usig a give data set to estimate the value of a fuctio or measuremet beyod the values already kow. F factor (ou) whole umber that is a divisor of aother umber; a algebraic expressio that is a divisor of aother algebraic expressio. is a factor of factor (verb) Fid the umber of algegbraic expressios that give a idicated product. To factor x x 6, write ( x )( x+ ). fractioal expoet expoet that is a ratioal umber. ( ) = = = frequecy table i a set of data. table that shows how ofte each item, umber, or rage of umbers occurs The data {,7, 6, 8, 9,,,,, 6,,, 0,, 9} ca be displayed as a frequecy distributio. Iterval Frequecy 6 7

13 fuctio f ( x ). rule that assigs to each umber x i the fuctio's domai a uique umber G geometric represetatio of the circular fuctios The represetatio of circular fuctios o a circle of uit radius. The trigoometric fuctios are called circular fuctios because their values are related to the legths of specific lie segmets associated with a circle of uit radius. θ O θ O D C O = cos θ, = si θ, O = CD = ta θ, OD = sec θ, OC = geometric sequece set of terms i which each term is formed by multiplyig the precedig term by a ozero costat. 8,,,,,... geometric series The idicated sum of a geometric sequece geometry rach of mathematics that deals with the properties, measuremet, ad relatioships of poits, lies, agles, surfaces, ad solids.

14 graphical represetatio graph or graphs used to model a mathematical relatioship. The figure below is a graphical represetatio of the locus of all poits uits from (0,7) ad uits from x = 8. graphical solutio of a system of equatios The set of poits i the plae whose coordiates are solutios to a system of equatios. greatest commo factor (GCF) The greatest umber or expressio that is a factor of two or more umbers or expressios. is the GCF of ad 6. xy is the GCF of x y ad 0xy growth factor The base of a expoetial fuctio. t I the formula f () t ab, = the base b, is the growth factor. H half-life The time eeded for a amout of a substace to decrease by oe-half. horizotal-lie test test usig ay horizotal lie to determie whether or ot a fuctio is oe-to-oe. fuctio f is oe-to-oe if ad oly if o horizotal lie itersects the graph of f more tha oce. I i The symbol represetig the basic uit of imagiary umbers, i =

15 idetities Equatios that are true for all values of the variables they cotai. x + = x + Si θ + Cos θ = image The resultig poit or set of poits uder a give trasformatio; i ay fuctio f, the image of x is the fuctioal value f ( x ) correspodig to x. RP,90 = ', the poit ' is the image of poit uder the rotatio, R P,90. I the fuctio f( x) = x +, 7 is the image of uder f. Examples: I trasformatioal geometry if ( ) imagiary umber imagiary uit. idex of a radical umber i the form bi, where b is a o-zero real umber ad i is the The letter i the expressio k is the idex i the expressio = iductive reasoig The process of observig data, recogizig patters ad makig geeralizatios about those patters. iterpolate The process of usig a give data set to estimate the value of a fuctio or measuremet betwee the values already kow. iterquartile rage resistat to outliers. iverse fuctio The differece betwee the first ad third quartiles; a measure of variability If the iverse of a fuctio is also a fuctio it is the a iverse fuctio. iverse of a fuctio The relatio formed whe the idepedet variable is exchaged with the depedet variable i a give relatio. iverse trigoometric fuctios Give the value of a trigoometric fuctio for a agle θ,the iverse trigoometric fuctio outputs the measure of agle θ. The iverses of the six basic trigoometric fuctios are writte as si -, cos -, ta -, csc -, sec -, ad cot -. iverse variatio relatioship i which the product of two variables is costat. iverse k variatio has a equatio i the form y =, where x ad y are variables ad k is a costat. x irratioal umber umber that caot be expressed as the ratio of two itegers. irratioal umber, i decimal form, is o-repeatig ad o-termiatig. The umbers π ad are irratioal umbers. J There are o J terms i the commecemet-level sectios.

16 6 K There are o K terms i the commecemet-level sectios. L Law of cosies I ay triagle, the square of oe side is equal to the sum of the squares of the other two sides dimiished by the product of those two sides ad the cosie of the icluded agle. Examples for Δ C: = + cos a b c bc = + cos b a c ac c a b ab C = + cos Law of sies I ay triagle the ratio of oe side to the sie of its opposite agle is equal to the ratio of ay other side ad the sie of its opposite agle. Examples for ay Δ C: a b c = = si si si C b c a C laws of expoets Rules ivolvig operatios o expressios with like bases. a b a b Multiplicatio law: x x = x + a b a b Divisio law: x x = x, x 0 a Power law: ( ) b ab x = x laws of logarithms The rules of logarithmic expressios havig like bases. Examples: Product rule: log = log + logb, b > 0 ad b b Quotiet rule: logb logb logb b > 0 =, ad b Power rule: lo g = log b, b > 0 ad b b b

17 least squares regressio lie The lie that fits data poits such that the sum of the squares of the vertical distaces betwee the predicted values o the lie ad the actual values is miimized. 7 liear equatio first degree equatio. Examples: y = 6x = + 7 liear regressio lie costructed usig the least-squares method. liear system set of two or more liear equatios with commo variables. logarithm log b a=. The expoet,, to which the base b must be raised to equal a, writte as log 8 = sice = 8. logarithmic form The expressio or a equatio cotaiig logarithms. The equatio log y = x is the logarithmic form of the expoetial equatio x = y M mea measure of cetral tedecy deoted by x, read x bar, that is calculated by addig the data values ad the dividig the sum by the umber of values. lso kow as the arithmetic mea or arithmetic average. measure of cetral agle The measure equal to the degree measure or radia measure of the arc itercepted by the agle. measure of cetral tedecy summary statistic that idicates the typical value or ceter of a orgaized data set. The three most commo measures of cetral tedecy are the mea, media, ad mode. measures of dispersio idicatio of the spread, or variatio, of data values about the mea. Some commo measures of dispersio are rage, quartiles, iterquartile rage, stadard deviatio ad variace. media measure of cetral tedecy that is, or idicates, the middle of a data set whe the data values are arraged i ascedig or descedig order. If there is o middle umber, the media is the average of the two middle umbers. Examples: The media of the set of umbers: {,,, 6, 7, 0, } is 6 The media of the set of umbers: {6, 7, 9, 0,, 7} is 9.

18 8 miute uit of degree measure equal to 60 of a degree. mode measure of cetral tedecy that is give by the data value(s) that occur(s) most frequetly i the data set. Examples: The mode of the set of umbers {, 6, 8, 6,,,, } is. The modes of the set of umbers {, 6, 7,,, 7, 9,,0} are ad 7. The mode of the set of umbers {0,, 7,,, } is oe or there is o mode. multiple represetatios Various ways, i.e., graphically, umerically, algebraically, geometrically, ad verbally, to preset, iterpret, commuicate, ad coect mathematical iformatio ad relatioships. N ature of the roots classificatio of the roots of a quadratic equatio, The discrimiat, b ac, idicates the ature of the roots of a quadratic equatio, ax + bx + c = 0 where a, b, ad c are ratioal umbers ad a 0. i.e. whether the roots are real or imagiary, ratioal or irratioal, equal or uequal. egative expoet expoet that is a egative umber. I geeral, x a = a, x 0. x ormal curve The graph of a ormal probability desity fuctio. This graph is bell-shaped ad symmetric about the mea ormal distributio distributio of data that varies about the mea i such a way that the graph of its probability desity fuctio is a ormal curve. The height of the curve is specified by the mea ad stadard deviatio of the distributio th root The solutio of x = c whe is odd or the oegative solutio of x = c whe is eve ad oegative. For ay real umber c ad ay positive iteger, the th root of c is deoted by either c or c. th term The fial term of a fiite sequece of elemets a, a,, a, or a arbitrary term of a ifiite sequece. O oe-to-oe fuctio fuctio where the iverse is also a fuctio.

19 9 oto fuctio mappi g, f : i which each elemet of set is the image of at least o e elemet i set. opposite side i a right triagle The side across from a agle. I a right triagle the hypoteuse is opposite the right agle ad each leg is opposite oe of the acute agles. With respect to, C is the opposite side, ad C is the adjacet side. With respect to side C, is the opposite agle. C ordiate The vertical coordiate of a two-dimesioal rectagular coordiate system; usually deoted by y. outlier data value that is far removed from the body of the data. Give the data set {,,, 6,, }, is the outlier. The value of the outlier will greatly effect o the value of the mea but ot the media. P parabola The locus of poits equidistat from a give poit (called the focus) ad a give lie (called the directrix). commo form of a equatio of a parabola with vertical lie symmetry is y = ax + bx + c, where a, b, ad c are real umbers ad a 0.

20 parameter quatity or costat whose value varies with the circumstaces of its applicatio. 0 I y = ax a is a parameter Pascal's Triagle triagular array of umbers composed of the coefficiets of ( x + y) where is ay whole umber. Each row begis ad eds with. The other values are foud by addig the two umbers that are above ad o either side of that value etc The coefficiets of ( x + y) are the umbers i the th row of Pascal s Triagle. ( x + y) = x + x y + 6x y + xy + y percetile score below which a certai percetage of the scores i a distributio fall. If a test score of 87 is the 7 th percetile of a distributio, the 7% of the scores are less tha 87 ad % of the scores are greater tha or equal to 87. period (of a fuctio) The horizotal distace after which the graph of a fuctio starts repeatig itself. The smallest value of k i a fuctio f for which there exists some costat k such that f () t = f( t+ k) for every umber t i the domai of f. periodic fuctio oscillatig fuctio that repeats its values at regular itervals; a fuctio f for which there exists some costat k such that f () t = f( t+ k) for every umber t i the domai of f. permutatio phase shift arragemet of objects i a specific order. The horizotal traslatio of a periodic graph. pi The irratioal umber equal to the legth of the circumferece of a circle divided by the legth of its diameter. polyomial expressio polyomial expressio i terms of x that ca be writte i the form ax + a x ax + ax+ a 0 where is a oegative iteger ad a is a real umber. polyomial fuctio fuctio that ca be writte i the form f x = a x + a x a x + a x+ a, where is a oegative iteger ad a is a real ( ) umber. 0 i i

21 powers of i The repetitive patter whe the imagiary uit, i, is raised to sequetial powers. i =, i = i, 6 7 i =, i = i, i =, i = i, etc. pricipal square root The positive square root of a umber. The pricipal square root of is. Pythagorea idetities The trigoometric idetities based o the Pythagorea Theorem. The idetity ad the idetities derived from it. Q Examples: x + = + ta x = sec + cot x = csc si cos x x quadratal agle agle i stadard positio whose termial side falls o a axis. 0,90,80,70,60 quadratic equatio equatio that ca be writte i the form ad c are real costats ad a = 0, where a, b, ax bx c quadratic formula The formula used to determie the roots of the quadratic b± b ac equatio ax + bx + c = 0 ; x =. a quadratic iequality secod degree iequality. x x 0 quartiles Values that divide a ordered data set ito fourths. The media, or secod quartile, divides the data ito a lower half ad a upper half; the first quartile Q is the media of the Q lower half; ad the third quartile Q is the media of the upper half R

22 radia agle measure i which oe full rotatio is π radias. Oe radia is the measure of a arc or the measure of the cetral agle tha itercepts that arc such that the arc s legth is the same as the radius of that circle. π radias = 80 degrees radia measure For a agle i stadard positio, the legth of the arc alog the uit circle from the poit (,0) o the iitial side to the poit P where the termial side itersects the uit circle. radical The root of a quatity as idicated by the radical sig. radical equatio radical expressio equatio that cotais at least oe term uder a radical sig. expressio that cotais at least oe term uder a radical sig. radical form The use of a radical sig to express a umber with a fractioal expoet. x = x radicad The quatity uder a radical sig; a umber or expressio from which a root is extracted. is the radicad of. I the expressio k, the radicad is k. radom sample sample i which all members of the populatio ad all groups of a give size have a equal chace of beig selected for the sample. rage (of a fuctio) The set of values of the depedet variable of a give fuctio; the set of secod coordiates i the ordered pairs of a fuctio. ratioal coefficiet coefficiet that is a ratioal umber. ratioal equatio equatio that cotais at least oe ratioal expressio. ratioal expressio The quotiet of two polyomials i the form, where ad are polyomials ad 0. x + x 9, x 9 0 ratioal iequality iequality which cotais a ratioal expressio. ratioal umber x x, x+ 0 x + y umber that ca be expressed as a ratio i the form a b where a ad b are itegers ad b 0. ratioal umber is either a termiatig or repeatig decimal.

23 ratioalize a deomiator a ratioal expressio. The process of chagig the irratioal deomiator of a fractio to + i + i + i = or = i + i reciprocal trigoometric fuctios The six fuctios: si x = csc x cos x = sec x ta x = cot x cot x = ta x sec x = cos x csc x = si x rectagular coordiates ordered pair of real umbers that establishes the locatio of a poit i a coordiate plae usig the distaces from two perpedicular itersectig lies called the coordiate axes. (See also Cartesia coordiates.) recursive rule For a sequece a, a, a,... a..., a formula that requires the computatio of all previous terms i order to fid the value of. a = a = a =... a = + a a referece agle The positive acute agle formed by the x-axis ad the termial side of a agleθ i stadard positio. regressio model fuctio (e.g., liear, expoetial, power, logarithmic) that fits a set of paired data. The model may eable other values of the depedet variable to be predicted. relatio correspodece betwee two sets; a set of ordered pairs

24 Examples: replacemet set setece. {(P,Q)} = {(,), (,0), (7,), (-,6)} The set whose members ca be substituted for the variable(s) i a ope restricted domai The domai resultig from a restrictio placed o a fuctio, based o the cotext of the problem. resultat The vector that is produced from the additio of two or more other vectors. root of a equatio solutio to a equatio of the form f(x) = 0. root of the equatio y = 6x 8 is because whe is substituted i for x, the value of y = 0. The roots of x x = 0 are x = ad x =. The equatio is true if we substitute either x = or x = ito the equatio. S sample space The set of all possible outcomes for a give evet. The sample space for tossig two cois is: {(H,H), (H,T), (T,H), (T,T)}. scatter plot graphical display of statistical data plotted as poits o a coordiate plae to show the correlatio betwee two quatities. elow is a table of paired quatitative data ad its scatter plot. Household Number of dogs Dollars spet each moth o dog food Perez 7 Joes 0 alcovich 0 Parso Motego Schwartz 7 0 arto 0 Walker 0 Dollars spet each moth o dog food Dog Food Expeses i 8 households $0 $0 $00 $80 $60 $0 $0 $ Number of Dogs I Household secat (of a agle) For a give acute agle θ i a right triagle, secθ, is the ratio of the legth of the hypoteuse to the legth of the side adjacet to the acute agle θ ; the reciprocal of the cosie ratio of the give agle. See also circular fuctio.

25 I this right triagle, sec = ad sec = C sector of a circle of the arc. regio bouded by a arc of the circle ad the two radii to the edpoits The shaded area i the circle below is a sector of circle O. sigma otatio ( ) shorthad way of writig a sum by usig the Greek letter sigma = = sie For a give acute agle θ i a right triagle, siθ, is the ratio of the legth of the side opposite the acute agle θ to the legth of the hypoteuse. See also circular fuctio. I this right triagle, si = ad si = C slope The measure of the steepess of a lie; the ratio of vertical chage to horizotal Δ y y chage; if poit P is (x,y ) ad poit Q is (x,y ) the slope of PQ is = y Δx x x.

26 7 9 = 8 The slope of the lie cotaiig the poits (-,7) ad (, -) is ( ) ( ) ( ) ( ) 6 solutio set y ad all value(s) of the variable(s) that satisfy a equatio, iequality, system of equatios, or system of iequalities. stadard deviatio (populatio) measure of variability. Stadard deviatio measures the average distace of a data elemet from the mea. If data is take from the etire populatio, divide by whe averagig the squared deviatios. The followig is the formula for populatio stadard deviatio: ( xi x) σ = stadard deviatio (sample) measure of variability. Stadard deviatio measures the average distace of a data elemet from the mea. If data is take from a sample istead of the etire populatio, divide by whe averagig the squared deviatios. The followig is the formula for sample stadard deviatio: s = ( xi x) stadard positio (of a agle) agle i the coordiate plae with its vertex at the origi ad its iitial side o the positive x-axis. subset set cosistig of elemets from a give set; it may be the empty set. if = {,,,,,6,7} ad = {,,}, the is a subset of. substitutio property y quatity ca be replaced by a equal quatity. If a+ x= b ad x = c the a+ c= b. subtractio property of equality If the same or equal quatities are subtracted from same or equal quatities, the the results are equal. If a = bthe a c = b c. sum ad product of roots of a quadratic equatio For a quadratic equatio b ax + bx + c = 0, a 0, whose roots are x ad x, the sum of the roots is x+ x =, ad the a c product of the roots is x x =. a sum of a fiite geometric series To determie the sum of the first terms of a geometric series, ( a r S ) =, r, where a is the first term ad r is the commo ratio. r

27 sum of a fiite arithmetic series To determie the sum of the first terms of a arithmetic series, S = ( a + a), where a is the first term, ad is the th a term. 7 sum or differece formulas for trigoometric fuctios trigoometric values of the sum or differece of two agles. Formulas used to determie ( ) ( ) si + = si cos + cos si cos + = cos cos si si ta ta + ta + = tata ( ) ( ) ( ) si = si cos cos si cos = cos cos + si si ta ta ta = + tata ( ) survey gatherig of facts or opiios by askig people questios through a iterview or questioaire. system of equatios/iequalities set of two or more equatios/iequalities. The solutio set cotais those values that satisfy all of the equatios/iequalities i the system. T taget (of a agle) For a give acute agle θ i a right triagle, taθ is the ratio of the legth of the side opposite the acute agle θ to the legth of the side adjacet to the agle θ. See also circular fuctio. I this right triagle, ta = ad ta = C termial side of a agle If side. C is the directed agle from to C the C is the termial

28 8 C theoretical probability The chaces of evets happeig as determied by calculatig results as they would occur uder ideal circumstaces. Whe you roll a die, oe of the six possible outcomes is a, so the theoretical probability of rollig a is. Three of the six possible outcomes are odd 6 umbers, so the theoretical probability of rollig a odd umber is 6. trasformatio oe-to-oe mappig of poits i the plae to poits i the plae. trasformatios of fuctios ad relatios of a trasformatio to a give fuctio. ew fuctio that results from the applicatio ( x ) Examples: The fuctio y = is a traslatio of the graph of y = x three uits to the right, while the fuctio y = ( x+ ) is a traslatio of the graph of y = x three uits to the left. trigoometric equatio equatio that uses variables expressed i terms of trigoometric fuctios. While x = x + is a algebraic equatio, siθ = siθ + is a trigoometric equatio. trigoometric fuctios The fuctios, sie, cosie, taget, cotaget, secat ad cosecat. U

29 udefied expressio i mathematics which does ot have meaig ad therefore is ot assiged a value. x + Whe x=, the expressio is udefied. x Whe x<, the expressio x is udefied i the Real umbers. Whe x<0, the expressio log x is udefied. 9 uit circle uivariate V variable variables. The circle of radius with ceter at the origi. set of data ivolvig oe variable. quatity whose value ca chage or vary; i algebra, letters ofte represet variace (populatio) measure of variability give by the average of squared deviatios. If data is take from the etire populatio, divide by whe averagig the squared deviatios. ( xi x) Populatio variace =. variace (sample) measure of variability give by the average of squared deviatios. If data is take from a sample istead of the etire populatio, divide by whe averagig the ( xi x) squared deviatios. Sample variace =. vector, quatity that has both magitude ad directio; represeted geometrically by a directed lie segmet. vertical lie test vertical lie draw to determie whether or ot a relatio is a fuctio. relatio is a fuctio if ad oly if o vertical lie itersects the graph of the relatio more tha oce. X, Y have o terms at this level. Z zero of a fuctio equal to 0. y value of the idepedet variable that makes the value of the fuctio zero product property If a ad b are real umbers, the ab = 0 if ad oly if a = 0 or b = 0, or a ad b = 0.