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1 Immcult Mthemtics Summer Assignment The purpose of summer ssignment is to help you keep previously lerned fcts fresh in your mind for use in your net course. Ecessive time spent reviewing t the beginning of clss prevents you from completing the work in the new course. Immcult is college prep school; you cnnot fford to fll behind nd not meet the cdemic epecttions of your colleges. Another importnt epecttion of colleges is tht students be independent lerners responsible for their own lerning while using the resources the colleges hve to offer. Your summer ssignment cn help you in meeting this epecttion s well if you tke your summer ssignment seriously. The work on the summer ssignment will not be collected or grded but you re responsible for the mteril it contins. College professors do not tke ecuses; neither do the mth techers t Immcult. 1. Downlod the summer ssignment for the course you will be tking in September, s well s the solutions document, from the Immcult website. Senior electives, Intro to Sttisticl Thinking nd Intro to Discrete Mthemtics, do not hve summer ssignment.. Work ll of the problems on loose lef or in notebook for future reference, checking your work s you go. If you cnnot do problem, consult your notebook from your previous mth course or go to n online source for n eplntion. Khn Acdemy is n ecellent online source of videos to eplin topics nd improve your understnding. Your work on the summer ssignment WILL NOT BE COLLECTED; however, you re responsible for its completion nd will be held responsible for its contents. If you need dditionl tutoring to be redy for your clss in September, you re responsible for securing it.. During the first full week of school, ech mth clss will provide one dy for nswering questions relted to the summer ssignment. Tutoring will lso be provided one dy fter school for ech subject if you need dditionl help. If you do not do the ssignment during the summer, you will hve no ide wht sorts of questions to sk during these sessions.. Unless n unepected schedule chnge occurs, ll mth students will hve n in-clss ssessment on the mteril of the summer ssignment during the first full week of school. The ssessment will be grded nd the grde WILL NOT BE SCALED. You will be more successful in your new course if you review during the summer. Good luck! Enjoy the summer. Immcult looks forwrd to seeing you in clss in September. Do not hesitte to e-mil the Mth Chir (epetsu@immculthighschool.org) if you hve ny questions bout the requirements of mth summer ssignments. All mth students re required to hve TI-8/8 grphing clcultor or comprble device. If you must purchse new TI-8/8, plese sve the "Proof of Purchse" crd found in the pckge; give it to your techer in September. These proofs of purchse cn be used to obtin free TI equipment for our school. Be Sprtn; be your best!!! Fith Scholrship Service Friendship

2 Nme: Algebr Techer: Answer Sheet for Intro to Preclc Summer Assignment nswer on pg nswer on pg nswer on pg nswer on pg Pge of 0

3 ORDER OF OPERATIONS Alwys follow the following order of opertions when solving ny mthemticl problem: Prentheses (grouping symbols) Eponent opertions Multiply or divide whichever comes first (left to right) Add or subtrct whichever comes first (left to right) Solve the following epressions. 1. (1 9) EVALUATING ALGEBRAIC EXPRESSIONS Substitute nd simplify fully. =, y =, z = y z 8. y z z y 9. y z 10. Pge of 0

4 SOLVING LINEAR EQUATIONS Concept: Solve liner equtions in one vrible. Remember: Wht is done to one side of the eqution MUST be done to the other side. Use the distributive property if the eqution contins prentheses. Combine ny like terms (sme vrible rised to sme power) on either side of the equl sign. Use ddition or subtrction to move terms or constnts round the equl sign. Use multipliction or division if the vrible hs coefficient. Check the solution. Solve the following liner equtions, showing ll work ( y ) n 5(n ) (n 1) 6 n Pge of 0

5 SOLVING LINEAR INEQUALITIES Concept: Solve liner inequlities, using sme process s for equtions. Remember: When you multiply or divide by negtive coefficient, you MUST chnge the inequlity sign to point the other wy: becomes, becomes, < becomes >, > becomes <. Solve the following liner inequlities, showing ll work. Grph its solution set d d Pge 5 of 0

6 SOLVING SYSTEMS OF LINEAR EQUATIONS There re wys to solve system of liner equtions: by substitution, nd liner combintion (which is lso clled elimintion). Substitution: 1. Solve one of the equtions for one of the vribles. Substitute the epression from Step 1 into the other eqution nd solve for the other vrible (you get number).. Substitute the number from step into the originl eqution used in step 1 nd solve.. Check the solution in ech of the originl equtions. Emple: + y=8 - y=-1 Solve for in nd eqution: = y 1 Substitute for in first eqution: (y-1)+y=7 Solve for y: y-+y=8 5y-=8 5y=10 y= Substitute y= into y = -1, = -1, =1 Check solution in both of the originl equtions. (1)+()=8 (yes), 1-=-1 (yes) Solve the following systems of equtions using the substitution method. 0. y y b 7 b. 7y y 9 Pge 6 of 0

7 Liner combintion / Elimintion: 1. Arrnge the equtions with like term in columns. Multiply one or both of the equtions by number to obtin coefficients tht re opposites for one of the vribles.. Add the equtions from step. Combining like terms will eliminte one vrible. Solve for the remining vrible.. Substitute the vlue obtined in step into either of the originl equtions nd solve for the other vrible. 5. Check the solution in ech of the originl equtions. 8 y 1 Emple: 5y 7 Multiply the second eqution by -. 8 y y 1 Add the equtions. Note the s re eliminted, leving only y. Solve for y. 7y 7 y 1 Substitute y=1 for y in either eqution, solve for. 8 (1) 1 8 Check solution in both equtions: 8(-)+(1)=-1 (yes) (-)+5(1)=-7 (yes) Solve the following systems of equtions using the liner combintion / elimintion method.. y 5 y 7. y 0 y 1 Pge 7 of 0

8 5. y y COORDINATE SYSTEM Concept: Grphing points in the coordinte plne ech point is represented by numbers: (,y), where the vlue tells you how mny spces to move long the -is, nd the y vlue tells you how mny spces to move up or down from the vlue. The result is point tht is grphed in the coordinte plne. The coordinte plne is divided into qudrnts, strting with the upper right (qudrnt I, (+,+)), moving counter-clockwise (qudrnt II, (-,+)), qudrnt III, (-,-)), nd qudrnt IV, (+,-)). Grph nd lbel the following points on the coordinte plne: 6. A (,5) 7. B (,-) 8. C (-,1) 9. D (-,-6) Pge 8 of 0

9 FINDING SLOPE GIVEN POINTS Slope is mesure of the rise nd run of n eqution of line. Given points, it cn be found by using the following formul: m rise run y y1 1 Find the slope of the line between the given points. 0. (7,) (5,) 1. (,-) (6,-5). (-,-) (0,0). (-,-5) (-,-) SLOPE AND Y-INTERCEPT Re-write the eqution, solving for y. The coefficient of represents the slope of the line, nd the constnt indictes the y-intercept, where the grph crosses the y-is. The generl formt is: y m b, where m (the coefficient of ) is slope nd b (the constnt) is the y-intercept. Rewrite the following equtions into slope-intercept form, nd stte the slope nd y- intercept.. y 5. y 6 6. y 1 7. y 16 Pge 9 of 0

10 SIMPLIFYING RADICALS Simplify rdicls by fctoring to find perfect squre numbers Simplifying gives ect nswer, using clcultor gives pproimtion Product rule: b b nd other wy: b b Quotient rule - but don t forget to simplify under the rdicl first: b b b nd other wy: b Rtionlizing denomintor - to rid rdicl from denomintor, multiply both top nd bottom by denomintor, nd simplify Simplify the following rdicl. Leve your nswer in simplest rdicl form. Do NOT use clcultor EXPONENT PROPERTIES Eponents re used to indicte powers. Their properties re listed below. Assume throughout your work tht no denomintor is equl to zero nd tht m nd n re positive integers. 0 1 m n mn ( m m b) b m ( m n mn ) n 1 n m n mn b n b n n Pge 10 of 0

11 Pge 11 of 0 Simplify ech epression. Use only positive integers for eponents. 6. ) )( ( y y y 51. t s r t s r 5. 6 y y 5. 0 k 5 h ADDING & SUBTRACTING POLYNOMIALS Like terms re defined s terms with the sme vrible rised to the sme power. When you hve like terms, you cn dd or subtrct them by dding or subtrcting the coefficients. Emple: +(-) = (+ -) = -1 or Add or subtrct the following polynomils by combining like terms

12 MULTIPLYING POLYNOMIALS Use the properties of eponents when multiplying vrible terms. You cn use distributive property or First-Inner-Outer-Lst methods. Emple - Distributive: Emple FOIL: ( 5)( ) ( ) 5( ) ( 5)( ) Specil Products: Recognizing when you cn multiply quickly by using one of the ptterns. Squre of sum or difference: b b b b b b Sum nd difference: b b b Multiply the following polynomils. 60. ( 8)( 9) 61. ( )( 9) 6. ( )( ) 6. ( 5 )( 7) 6. ( 9) 65. ( ) 66. 5) ( 68. b b 69. Pge 1 of 0

13 FACTORING Gretest common fctor Finding the vrible or number tht is common, nd creting distributive problem. Emple: 8 6 ( ) Find the gretest common fctor in ech problem y y 8y 10y Fctoring Trinomils (generl nd ptterns) Think bckwrds multipliction of binomil fctors. 6 8 Wht did you multiply to get? Wht did you multiply to get 8? Signs need to be +, + Wht fctors of 8 cn you dd together to get 6? (+)(+) Check with FOIL: Ptterns squre of sum, nd squre of difference, nd sum nd difference pttern. Key recognizing when you cn fctor using specil products. Recognize the pttern nd the fctors tht were multiplied to get the pttern. b b ( b) b b b ( b ) ( b)( b) Pge 1 of 0

14 Fctor the following polynomils y 0y Pge 1 of 0

15 ANGLE MEASURES, PARALLEL & PERPENDICULAR LINES Verticl ngles re lwys congruent. If two lines re perpendiculr ( ), tht mens tht ll of the ngles formed by their intersection re 90. If two lines re prllel ( ), tht mens tht the lines never intersect. When two prllel lines re cut by trnsversl: o Alternte interior ngles re congruent o Cooresponding ngles re congruent o Consecutive interior ngles re supplementry Using the digrm below, find ll of the requested ngle mesures. In the digrm below, AD BC nd AB AD. Also, m CDA 7. 1 A D B 5 C m 1 8. m 85. m m m m m m 6 Pge 15 of 0

16 PYTHAGOREAN THEOREM Pythgoren Theorem sys b c, where c is the length of the hypotenuse of right tringle nd nd b re the lengths of the legs. Find the length of the missing side of the right tringle using Pythgoren Theorem. Leve your nswer in simplest rdicl form km km 0 cm 1 cm in 5 ft 5 ft 6 in CIRCLES & TANGENT LINES Lines re tngent to circle if they intersect the circle t only one point. If line is tngent to circle, then it is perpendiculr to the rdius drwn to the point of tngency. In the problems below, ssume tht lines tht pper to be tngent re tngent. 95. CA = 15 ft, AB = 9 ft. Find CB. 96. GJ = 9, HJ = 1. Find GF. C H J G D A B F Pge 16 of 0

17 AREAS OF BASIC FIGURES Are formuls for bsic shpes re below. You re encourged to memorize these formuls. Remember tht the height of tringle/prllelogrm/trpezoid is NOT necessrily the length of one of its sides, but n ltitude drwn to the bse of the figure. Remember tht the rdius of circle is one-hlf the length of its dimeter. Remember units! Tringle: 1 A bh Circles: A r Regulr Polygon: 1 A P Rectngle: A lw Prllelogrm: A bh Squres: A s Trpezoid: 1 A h( b 1 b ) Find the re of ech of the figures drwn below. Leve nswers in terms of π in 11 mm ft 1 ft cm 7 cm cm Pge 17 of 0

18 VOLUME & SURFACE AREA OF D SOLIDS Surfce Are is the sum of the res of ll the sides of D solid. To find the surfce re of prism, dd up ll of its sides. To find the surfce re of cylinder, multiply the circumference of the bse with the height of the cylinder, nd then dd this to the res of the two bses. Volume is how much spce is inside D solid. Volume of prism/cylinder = (Are of one Bse)*(Height of the Prism/Cylinder) Volume of cone/pyrmid = (1/)*(Are of the Bse)*(Height of the Cone/Pyrmid) Emple: Find the surfce re & volume of the cylinder below. cm SURFACE AREA: Circumference = πr = π* = 8π cm Circumference * Height = 80π cm Are of ech Bse = πr = 16π cm 10 cm Surfce Are = 80π + 16π + 16π = 11π cm VOLUME: (Are of one Bse)*(Height) = (πr )*(height) 16π*10 = 160π cm Emple: Find the surfce re & volume of the prism below. SURFACE AREA: Are of one Tringle = (1/)*(Bse)*(Height) = (1/)** = 6 ft Are of Bottom Side = *7 = 8 ft ft Are of Verticl Side = *7 = 1 ft ft 7 ft Are of Slnted Side = 5*7 = 5 ft (Use Pythgoren Thm) Surfce Are = = 1 ft VOLUME: Volume = (Are of one Bse)*(Height) = (6 ft )*(7 ft) = ft Pge 18 of 0

19 Find the surfce re nd volume of the following solids. Leve nswers in terms of π m 6 m 9 in in in LAW OF SINES & LAW OF COSINES When you re missing informtion for n cute or obtuse tringle, rther thn use SOHCAHTOA, you need to use Lw of Sines or Lw of Cosines. Lw of Cosines is useful for SAS nd SSS. Lw of Sines is useful for ASA nd AAS. Lw of Sines my be used for SSA s well, but since SSA does not prove tringle congruence, you my hve to solve for more thn one tringle! A c b B C Lw of Sines: If hs sides of length, b, nd c, which re opposite from ngles A, B nd C respectively (s shown in the figure), then Lw of Cosines: If hs sides of length, b, nd c, which re opposite from ngles A, B nd C respectively (s shown in the figure), then: Pge 19 of 0

20 Use Lw of Sines to solve the following problems. Round your nswers to the nerest thousndth. HINT: Drw the tringle first! 10. Find c if A = 8, B = 85, nd b = Find b if C = 8, = 10, nd B = 71 Use Lw of Cosines to solve the following problems. Round your nswers to the nerest thousndth. HINT: Drw the tringle first! 105. Find b if = 5, c =, nd B = 88 Pge 0 of 0