GRAPHING LINEAR EQUATIONS. Linear Equations ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.

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1 GRAPHING LINEAR EQUATIONS Quadrat II Quadrat I ORDERED PAIR: The first umer i the ordered pair is the -coordiate ad the secod umer i the ordered pair is the y-coordiate. (,1 ) Origi ( 0, 0 ) _-ais Liear Equatios Quadrat III Quadrat IV Slope = chage i y chage i Lies l 1 ad l are parallel to each other. The slope of l 1 is the same as the slope of l. Lies l 1 ad l are perpedicular to l. The slope of l 1 is the Negative Reciprocal of the slope of l. Give two poits o a lie ( 1,y 1 ) ad (,y ) The slope ca e foud through the equatio m = y -y 1 1 The slope-itercept form of a liear equatio is y = m +, where m is the slope ad ( 0, ) is the y-itercept. The poit slope formula for a lie with poit ( 1, y 1 ) ad slope m is y y 1 = m( 1 ) Slope of l 1 = / Slope of l = -¾ y-ais l 1 y-itercept of l 1 is ( 0, ) To fid the y-itercept, set =0 ad solve for y -itercept of l 1 is ( -, 0 ) To fid the -itercept, set y =0 ad solve for. l l Equatio for l 1 y = + Equatio for l is. Use poit (6. -1) ad slope / i poit-slope formula: y ( 1) = y + 1 = y = 9 8 Slope of l = / ( 6 ) Equatio for l is y = 5 Equatio for this horizotal lie is y= This lie has a slope of 0 Equatio for this vertical lie is = This lie has NO SLOPE

LINEAR INEQUALITIES To graph a LINEAR INEQUALITY, First rewrite the iequality to solve for y. If the resultig iequality is y >., The make a dashed lie ad shade the area ABOVE the lie.

2 LINEAR INEQUALITIES To graph a LINEAR INEQUALITY, First rewrite the iequality to solve for y. If the resultig iequality is y >., The make a dashed lie ad shade the area ABOVE the lie. If the resultig iequality is y <.., The make a dashed lie ad shade the area BELOW the lie. If the resultig iequality is y.., The make a SOLID lie ad shade the area ABOVE the lie. If the resultig iequality is y., The make a SOLID lie ad shade the area BELOW the lie. If whe isolatig y, you must divide oth sides of iequality y a egative umer, The the iequality sig must e SWITCHED. Eample: -y < 1 y > Graph the solutio set of y 1 y 1 y + 1 Switch iequality sig whe dividig y a egative umer. y y 6 + 1

3 Iput FUNCTIONS Iput Fuctio f() Output y=f() Fuctio f()=-1 Output f() = () 1=5 A fuctio, f, is like a machie that receives as iput a umer,, from the domai, maipulates it, ad outputs the value, y. The fuctio is simply the process that goes through to ecome y. This machie has restrictios: 1. It oly accepts umers from the domai of the fuctio.. For each iput, there is eactly oe output (which may e repeated for differet iputs). OFFICIAL DEFINITION OF A FUNCTION: Let ad Y e two oempty sets. A fuctio from ito Y is a relatio that associates with each elemet of X, eactly _oe elemet of Y. However, a elemet of Y may have more tha oe elemets of associated with it. That is, for each ordered pair (,y), there is eactly oe y value for each, ut there may e multiple -values for each y. The variale is called the idepedet variale (also sometimes called the argumet of the fuctio), ad the variale y is called depedet variale (also sometimes called the image of the fuctio.) Aalogy: I the -y relatio -ship, the s are the wives ad the y s are the husads. A husad is allowed to have more tha oe wife, ut each wife() is oly allowed 1 husad(y). A relatio is a correspodece etwee two sets. If ad y are two elemets i these sets ad if a relatio eists etwee ad y, the correspods to y, or y depeds o. Hours Studyig i Math La y Score o Math Test The set of -coordiates {,,,,5,6,6,7} correspods to the set of y coordiates {60,70,70,80,85,85,95,90} The set of distict -coordiates is called the _DOMAIN of the relatio. This is the set of all possile values specified for a give relatio. The set of all distict y values correspodig to the - coordiates is called the RANGE. I the eample aove, Domai = {,,,5,6,7} Rage = {60,70,80,85,85,95,90} This relatio is ot a fuctio ecause there are two differet y-coordiates for the -coordiate,, ad also for the -coordiate, 6.

4 Suppose a maufacturer has determied that at a price of $115, cosumers will purchase 1,000,000 portale C players ad that at a price of $90, cosumers will purchase 1,50,000 portale CD players. This data ca e w i tale format, where the iput represets the price of CD player ad the output represets the umer of CD players. Price of CD players, Numer of CD players, y $115 1,000,000 $90 1,50,000 (a) Plot the two poits o the grid elow ad sketch the lie cotaiig them. E the lie so that it itersects the vertical ais. Numer of CD players,500,000,00,000,00,000,00,000,100,000,000,000 1,900,000 1,800,000 1,700,000 1,600,000 1,500,000 1,00,000 1,00,000 1,00,000 1,100,000 1,000, , , , , ,000 00,000 00,000 00, ,000 0 $0 $5 $10 $15 $0 $5 $0 $5 $0 $5 $50 $55 $60 $65 $70 $75 $80 $85 $90 $95 $100 $105 $110 $115 $10 $15 Price of CD players ($) () Determie the slope of the lie. What are its uits of measuremet? What is practical meaig of the slope i this situatio? (c) Write a equatio for the lie cotaiig the two poits you graphed. (d) What is the vertical itercept of the lie you graphed ad whose equatio yo wrote? What is the practical meaig of the vertical itercept i this situatio Does this make ay sese? (e) Use this fuctio to predict how may portale CD players cosumers will purchase if the price is $80. Write your aswer as a complete setece. (f) Use this model to determie the price of CD players if the cosumers purcha 1.80 millio CD players. Write your aswer as a complete setece.

5 SYSTEMS OF LINEAR EQUATIONS A system of liear equatios is a set of two equatios of lies. A solutio of a system of liear equatios is the set of ordered pairs that makes each equatio true. That is, the set of ordered pairs where the two lies itersect. If the system is INDEPENDENT, there is ONE SOLUTION, a ordered pair (,y) If the system is DEPENDENT, there is INFINITELY MANY SOLUTIONS, all (,y) s that make either equatio true (sice oth equatios are essetially the same i this case. If the system is INCONSISTENT, there are NO SOLUTIONS, ecause the two equatios represet parallel lies, which ever itersect. GRAPHING METHOD. Graph each lie. This is easily 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 equatio to isolate a variale to solve for. I this system, solvig for y i the secod equatio makes the most sese, sice y is already positive ad has a coefficiet of 1. This secod equatio turs ito y = Now that you have a equatio for y i terms of, sustitute that equatio for y i the first equatio i your system. Sustitute y = i y =5 (- + 5) = 5 Simplify ad solve for. + 5 = = 5 5 = 10 = ADDITION METHOD 5 + y = y = Elimiate oe variale y fidig the LCM of the coefficiets, the multiply oth sides of the equatios y whatever it takes to get the LCM i oe equatio ad LCM i the other equatio. After this we ca add oth equatios together ad elimiate a variale. Let s choose to elimiate y. The y terms are y ad -7y. The LCM is 1. Multiply the first equatio y 7 ad the secod equatio y. 7(5 + y) = -9(7) (1 7y) = () 5 + 1y = -6 1y = Add them together 59 = -59 = -1 Solve for y y sustitutig = ito y = y = -() + 5 = = -1 Therefore solutio is (,-1) CHECK y sustitutig solutio ito the other equatio ad see if it is true. + y = 5 () + (-1) =5 5 = 5 YES! Solve for y y sustitutig =-1 ito either of the two origial equatios. 5(-1) + y = -9 y = - y = - Therefore solutio is (-1,-) CHECK y sustitutig solutio ito the other equatio ad see if it is true. 1 7y = 1(-1) 7(-) = -1+1 = = YES!

6 DEPENDENT SYSTEM INDEPENDENT SYSTEM (ONE SOLUTION) INCONSISTENT SYSTEM (NO SOLUTION) (INFINITELY MANY SOLUTIONS) Y=X+ Y=-X+5 Y=-.5X + Y=-.5X + Y=X-1 Y=X-1 10 X 6 Y=-.5X + Y=-.5X + 0 X Y=X-1 Y=X X Y=X+ Y=-X , SYSTEM: SYSTEM: SYSTEM: - y = - + y = y = + y = 5 + y = y = Graphig Method Graphig Method Graphig Method Chage each equatio so y is isolated. Chage each equatio so y is isolated. Chage each equatio so y is isolated. + y = y = - y = - y = y = y = - - y = -½ + y = - 1 y = + Graph this lie. Y-itercept is (0,) Slope = -½ Graph this lie. Y-itercept is (0,) Slope = 1 + y = y = + y = 5 y = y = -9 + y = y = -½ + y = - 1 Graph this lie. Y-itercept is (0,5) Slope = - Graph this lie. Y-itercept is (0,) Slope = -½ Itersectio poit is the solutio Lies are parallel (have the same slope ut differet y-itercepts) Both equatios are the lie whose Solutio: (1,) No Iterceptio Poit, therefore NO SOLUTION y-itercept is (0,-1) ad slope = Therefore, there are INFINITELY MANY SO Susitutio Method Susitutio Method Susitutio Method - y = - + y = y = + y = 5 + y = y = Choose oe of the equatios to get a variale y itself Choose oe of the equatios to get a variale y itself Choose oe of the equatios to get a variale - y = - + y = y = = y - = -y + 8 -y = -6 + Use that epressio to sustitute ito the other equatio Use that epressio to sustitute ito the other equatio y = -1 + y = 5 (-y + 8) + y = 8 Use that epressio to sustitute ito the othe (y - ) + y = 5 -y y = y = Solve for that variale Solve for that variale 9 - ( - 1) = y - + y = = 8 Solve for that variale y = 9 No variales left, ad you have a false statemet = y = Therefore, NO SOLUTION 0 + = Now use that value to solve for i the first equatio. No variales left, ad you have a true stateme = y - Therefore, INFINITELY MANY SOLUTIONS = - = 1 Solutio: (1,) Elimiatio Method: Elimiatio Method: Elimiatio Method: - y = - + y = y = + y = 5 + y = y = Choose a variale to elimiate. Y would e easiest. Choose a variale to elimiate. would e easiest. Choose a variale to elimiate. I'll choose y. LCM of coefficiets is 1. Goal: 1, -1. Already there! LCM of coefficiets is. Goal: -, as ew coefficiets LCM of coefficiets of y is 6. Goal: 6, -6 as e Add equatios together. -( + y) = 8(-) Multiply oth sides of first equatio y - ad = + y = 8 multiply oth sides of secod equatio y = 1 - -y =-16 Solve for y i oe of the equatios. -(6 - y) = (-) + y = 5 Add equatios together. (9 - y) = () (1) + y =5 0 = -8 y = 5- No variales left, ad a false statemet y = -6

7 WORD PROBLEMS Settig up word prolems: 1) Fid out what you are eig asked to fid. Set a variale to this ukow quatity. Make sure you kow the uits of this ukow (miles?, hours? ouces?) ) If there is aother ukow quatity, use the give iformatio to put that ukow quatity i terms of the variale you have chose. If ot eough iformatio is give, use aother variale. ) Set up a tale with a row for each ukow ad colums made up of the terms of oe of equatios aove (Rate*Time=Distace, Amt*Uit Cost=Value, etc..) ) Use the give iformatio to comie the equatios of each row of the tale ito equatios. You eed at least oe equatio for every variale you eed to solve for. (Oe variale eeds 1 equatio, Two variales eeds two equatios, etc..) 5) Oce oe variale is solved for, you ca fid the other ukow. 6) Check your equatio y pluggig i your value for ad seeig if your equatio is true. The state your coclusio i a complete setece.

8 Eample: A seaplae flyig with thewid flew from a ocea port to a lake, a istace of 0 miles, i hours. Flyig agaist the wid, it make the trip from the lake to the ocea port i hours. Fid the rate of the plae i calm air ad the rate of the wid. Step 1) What are we asked to fid? The rate of the plae i calm air ad the rate of the wid. (Miles per Hour) ) Let p = rate of the plae i calm air ad w = rate of the wid. ) The trip to the lake ad ack are the SAME DISTANCE. So oth rows are equal to the distace give, 0 miles. Rememer that whe you fly with the wid, you actual rate is the rate of the plae PLUS the rate of the wid. Whe you fly agaist the wid, your actual rate is the rate of the plae MINUS the rate of the wid. Tale: ) Equatios from the tale. p + w = 0 Rate Time Distace = rate*time With the wid p+w hrs 0 miles Agaist the wid p-w hrs 0 miles p w = 0 Solve usig additio method. Pick a variale to elimiate the multiply each equatio ut the umer ecessary to get OPPOSITE LCMs of the coefficiets of that variale. If we elimiate w, the LCM of w ad -w is 6. So our goal is to have w s coefficiets to e 6 ad -6. (p + w) = (0) 6p + 6w = 70 (p w) = (0) 6p - 6w= 80 1p = 100 p = 100 5) Solve for the other ukow usig oe of the equatios. (100) + w = w = 0 w = 0 w =0 6) Check p=100 ad w = 0 usig the other equatio to see if you are correct. (100) (0) =? = 0 yes CONCLUSION: The rate of the plae i calm air is 100mph ad the rate of the wid is 0mph

9 EXPONENTS & POLYNOMIALS Epoet - A umer or symol, as i ( + y), placed to the right of ad aove aother umer, variale, or epressio (called the ase), deotig the power to which the ase is to e raised. Also called power. The epoet (or power) tells how may times the ase is to e multiplied y itself. Eample 1: ( + y) = ( + y)(+y)(+y) Eample : (-) = (-)(-)(-)(-) = 81 Properties of Epoets If m ad are itegers, the m m+ = If m ad are itegers, the m m ( ) = If is a real umer ad 0, the 0 = 1 If m,, ad p are itegers, the ( y) = y, ad ( y ) = y m p m p p Eample: a a 8 1 a = a 1 8 If m ad are itegers ad 0, the = a m = m If is a positive iteger (- is egative), ad 0, the = 1 1 ad = If is a iteger ad 0, the a a = If is a positive iteger (- is egative), ad 0, the a = a ( 1) 8 ( ) Dividig Polyomials ( ) () = = + ( + ) (-1)= - ( ) ( - ) 0 = a 1 6 = a 6 POLYNOMIALS polyomial is a term or sum of terms i which all variales have whole umer epoets. Eample:, or + 1, or moomial a umer, a variale, or a product of umers ad variales. Eample:,, - are all moomials. iomial the sum of two moomials that are ulike terms. triomial the sum of three moomials that are ulike terms. like terms terms of a variale epressio that have the same variale ad the same epoet. degree highest epoet of a variale i a polyomial. FOIL method A method of fidig the product of two iomials i which the sum of the products of the First terms, of the Outer terms, of the Ier terms, ad of the Last terms is foud. Eample: (+)(+) = * = Bo Method Perfect-Square Triomials (a+) = a + a + (+) = Differece of squares a - = (a+)(a-) 5 = ( + 5)( 5) Rules for Variale Epressios: Oly like terms ca e added, ad whe addig like terms, do ot chage the epoet of the variale. 5 + = 8 Whe multiplyig variale epressios, add epoets of like variales (5y )(y )=10y + = 10y 5 Whe takig powers of variale epressio that is a moomial (oe term), multiply epoets of EVERY term iside the paretheses ( y ) = * y * = 8 9 y 1

GRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.

GRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1. 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