GRAPHING LINEAR EQUATIONS Qudrt II Qudrt I ORDERED PAIR: The first umer i the ordered pir is the -coordite d the secod umer i the ordered pir is the y-coordite. (, ) Origi ( 0, 0 ) _-is Lier Equtios Qudrt III Qudrt IV Slope = chge i y chge i Lies l d l re prllel to ech other. The slope of l is the sme s the slope of l. Lies l d l re perpediculr to l. The slope of l is the Negtive Reciprocl of the slope of l. Give two poits o lie (,y ) d (,y ) The slope c e foud through the equtio m = y -y The slope-itercept form of lier equtio is y = m +, where m is the slope d ( 0, ) is the y-itercept. The poit slope formul for lie with poit (, y ) d slope m is y y = m( ) Slope of l = / Slope of l = -¾ y-is l y-itercept of l is ( 0, ) To fid the y-itercept, set =0 d solve for y -itercept of l is ( -, 0 ) To fid the -itercept, set y =0 d solve for. l l - - - - - - - Equtio for l y Equtio for l is. Use poit (6. -) d slope / i poit-slope formul: y y y ( ) ( 9 Slope of l = / 8 Equtio for this horizotl lie is y= This lie hs slope of 0 6) Equtio for this verticl lie is = This lie hs NO SLOPE Equtio for l is y 5
LINEAR INEQUALITIES To grph LINEAR INEQUALITY, First rewrite the iequlity to solve for y. If the resultig iequlity is y >., The mke dshed lie d shde the re ABOVE the lie. If the resultig iequlity is y <.., The mke dshed lie d shde the re BELOW the lie. If the resultig iequlity is y.., The mke SOLID lie d shde the re ABOVE the lie. If the resultig iequlity is y., The mke SOLID lie d shde the re BELOW the lie. If whe isoltig y, you must divide oth sides of iequlity y egtive umer, The the iequlity sig must e SWITCHED. Emple: -y < y > y y Grph the solutio set of y Switch iequlity sig whe dividig y egtive umer. y y 6
Hours Studyig i Mth L Iput FUNCTIONS Iput Fuctio f() Output y=f() Fuctio f()=- Output f() = A fuctio, f, is like mchie tht receives s iput umer,, from the domi, mipultes it, d outputs the vlue, y. The fuctio is simply the process tht goes through to ecome y. This mchie hs restrictios:. It oly ccepts umers from the domi of the fuctio.. For ech iput, there is ectly oe output (which my e repeted for differet iputs). OFFICIAL DEFINITION OF A FUNCTION: Let d Y e two oempty sets. A fuctio from ito Y is reltio tht ssocites with ech elemet of X, ectly _oe elemet of Y. However, elemet of Y my hve more th oe elemets of ssocited with it. Tht is, for ech ordered pir (,y), there is ectly oe y vlue for ech, ut there my e multiple -vlues for ech y. The vrile is clled the idepedet vrile (lso sometimes clled the rgumet of the fuctio), d the vrile y is clled depedet vrile (lso sometimes clled the imge of the fuctio.) Alogy: I the -y reltio -ship, the s re the wives d the y s re the husds. A husd is llowed to hve more th oe wife, ut ech wife() is oly llowed husd(y). A reltio is correspodece etwee two sets. If d y re two elemets i these sets d if reltio eists etwee d y, the correspods to y, or y depeds o. Hours Studyig i Mth L 5 6 6 7 y Score o Mth Test 60 70 70 80 85 85 95 90 Score o Mth Test 00 80 60 0 0 0 0 6 8 Score o Mth Test The set of -coordites {,,,,5,6,6,7} correspods to the set of y coordites {60,70,70,80,85,85,95,90} The set of distict -coordites is clled the _DOMAIN of the reltio. This is the set of ll possile vlues specified for give reltio. The set of ll distict y vlues correspodig to the - coordites is clled the RANGE. I the emple ove, Domi = {,,,5,6,7} Rge = {60,70,80,85,85,95,90} This reltio is ot fuctio ecuse there re two differet y-coordites for the -coordite,, d lso for the -coordite, 6.
SYSTEMS OF LINEAR EQUATIONS A system of lier equtios is set of two equtios of lies. A solutio of system of lier equtios is the set of ordered pirs tht mkes ech equtio true. Tht is, the set of ordered pirs where the two lies itersect. If the system is INDEPENDENT, there is ONE SOLUTION, ordered pir (,y) If the system is DEPENDENT, there is INFINITELY MANY SOLUTIONS, ll (,y) s tht mke either equtio true (sice oth equtios re essetilly the sme i this cse. If the system is INCONSISTENT, there re NO SOLUTIONS, ecuse the two equtios represet prllel lies, which ever itersect. GRAPHING METHOD. Grph ech lie. This is esily doe e puttig them i slope-itercept form, y = m +. The solutio is the poit where the two lies itersect. SUBSTITUTION METHOD. y = 5 + y = 5 Choose equtio to isolte vrile to solve for. I this system, solvig for y i the secod equtio mkes the most sese, sice y is lredy positive d hs coefficiet of. This secod equtio turs ito y = - + 5 Now tht you hve equtio for y i terms of, sustitute tht equtio for y i the first equtio i your system. Sustitute y = - + 5 i y =5 (- + 5) = 5 Simplify d solve for. + 5 = 5 5 5 = 5 5 = 0 = ADDITION METHOD 5 + y = -9 7y = Elimite oe vrile y fidig the LCM of the coefficiets, the multiply oth sides of the equtios y whtever it tkes to get the LCM i oe equtio d LCM i the other equtio. After this we c dd oth equtios together d elimite vrile. Let s choose to elimite y. The y terms re y d -7y. The LCM is. Multiply the first equtio y 7 d the secod equtio y. 7(5 + y) = -9(7) ( 7y) = () 5 + y = -6 y = Add them together 59 = -59 = - Solve for y y sustitutig = ito y = - + 5. y = -() + 5 = -6 + 5 = - Therefore solutio is (,-) CHECK y sustitutig solutio ito the other equtio d see if it is true. + y = 5 () + (-) =5 5 = 5 YES! Solve for y y sustitutig =- ito either of the two origil equtios. 5(-) + y = -9 y = - y = - Therefore solutio is (-,-) CHECK y sustitutig solutio ito the other equtio d see if it is true. 7y = (-) 7(-) = -+ = = YES!
WORD PROBLEMS Settig up word prolems: ) Fid out wht you re eig sked to fid. Set vrile to this ukow qutity. Mke sure you kow the uits of this ukow (miles?, hours? ouces?) ) If there is other ukow qutity, use the give iformtio to put tht ukow qutity i terms of the vrile you hve chose. If ot eough iformtio is give, use other vrile. ) Set up tle with row for ech ukow d colums mde up of the terms of oe of equtios ove (Rte*Time=Distce, Amt*Uit Cost=Vlue, etc..) ) Use the give iformtio to comie the equtios of ech row of the tle ito equtios. You eed t lest oe equtio for every vrile you eed to solve for. (Oe vrile eeds equtio, Two vriles eeds two equtios, etc..) 5) Oce oe vrile is solved for, you c fid the other ukow. 6) Check your equtio y pluggig i your vlue for d seeig if your equtio is true. The stte your coclusio i complete setece.
Emple: A seple flyig with thewid flew from oce port to lke, istce of 0 miles, i hours. Flyig gist the wid, it mke the trip from the lke to the oce port i hours. Fid the rte of the ple i clm ir d the rte of the wid. Step ) Wht re we sked to fid? The rte of the ple i clm ir d the rte of the wid. (Miles per Hour) ) Let p = rte of the ple i clm ir d w = rte of the wid. ) The trip to the lke d ck re the SAME DISTANCE. So oth rows re equl to the distce give, 0 miles. Rememer tht whe you fly with the wid, you ctul rte is the rte of the ple PLUS the rte of the wid. Whe you fly gist the wid, your ctul rte is the rte of the ple MINUS the rte of the wid. Tle: ) Equtios from the tle. p + w = 0 Rte Time Distce = rte*time With the wid p+w hrs 0 miles Agist the wid p-w hrs 0 miles p w = 0 Solve usig dditio method. Pick vrile to elimite the multiply ech equtio ut the umer ecessry to get OPPOSITE LCMs of the coefficiets of tht vrile. If we elimite w, the LCM of w d -w is 6. So our gol is to hve w s coefficiets to e 6 d -6. (p + w) = (0) 6p + 6w = 70 (p w) = (0) 6p - 6w= 80 p = 00 p = 00 5) Solve for the other ukow usig oe of the equtios. (00) + w = 0 00 + w = 0 w = 0 w =0 6) Check p=00 d w = 0 usig the other equtio to see if you re correct. (00) (0) =? 0 00 60 = 0 yes CONCLUSION: The rte of the ple i clm ir is 00mph d the rte of the wid is 0mph
EXPONENTS & POLYNOMIALS Epoet - A umer or symol, s i ( + y), plced to the right of d ove other umer, vrile, or epressio (clled the se), deotig the power to which the se is to e rised. Also clled power. The epoet (or power) tells how my times the se is to e multiplied y itself. Emple : ( + y) = ( + y)(+y)(+y) Emple : (-) = (-)(-)(-)(-) = 8 Properties of Epoets If m d re itegers, the m m If m d re itegers, the m m ( ) If is rel umer d 0, the 0 If m,, d p re itegers, the ( y) y, d m p m p p ( y ) y Emple: 8 Dividig Polyomils ( + ) () = 8 6 If m d re itegers d 0, the m m If is positive iteger (- is egtive), d d If is iteger d 0, the If is positive iteger (- is egtive), d ( ) 8 ( ) 0, the 0, the 6 6 POLYNOMIALS polyomil is term or sum of terms i which ll vriles hve whole umer epoets. Emple:, or +, or - + + moomil umer, vrile, or product of umers d vriles. Emple:,, - re ll moomils. iomil the sum of two moomils tht re ulike terms. triomil the sum of three moomils tht re ulike terms. like terms terms of vrile epressio tht hve the sme vrile d the sme epoet. FOIL method A method of fidig the product of two iomils i which the sum of the products of the First terms, of the Outer terms, of the Ier terms, d of the Lst terms is foud. Emple: (+)(+) = + + + * = + 5 + 6 commo fctor fctor tht is commo to two or more umers. Emple: Wht re the commo fctors of d 6? The fctors of re,,,,6,, d The fctors of 6 re,,,8,6,, The commo fctors of d 6 re,, d The Gretest Commo Fctor of d 6 is. Perfect-Squre Triomils (+) = + + (+) = + 6 + 9 Differece of squres - = (+)(-) 5 = ( + 5)( 5) Rules for Vrile Epressios: Oly like terms c e dded, d whe ddig like terms, do ot chge the epoet of the vrile. 5 + = 8 ( + ) (-)= Whe multiplyig vrile epressios, dd epoets of like vriles (5y )(y )=0y + = 0y 5 - - - - Whe tkig powers of vrile epressio tht is moomil (oe term), multiply epoets of EVERY term iside the pretheses. 0