1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE


 Claude Gibson
 9 months ago
 Views:
Transcription
1 ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check your work nd look for mistkes. If you hve troule, sk mth techer or someone else who understnds this topic. TOPIC : ARITHMETIC OPERATIONS A. Frctions: Simplifying frctions: exmple: Reduce 7 6 : (Note tht you must e le to find common fctor; in this cse 9; in oth the top nd the ottom in order to reduce.) Prolems : Reduce: Equivlent frctions: exmple: is equivlent to how mny eighths? Prolems : Complete: How to get the lowest common denomintor (LCD) y finding the lest common multiple (LCM) of ll denomintors: exmple: 6 nd 8. First find LCM of 6 nd : 6 LCM 0, so 6 0, nd Prolems 67: Find equivlent frctions with the LCD: 6. nd nd 7 8. Which is lrger, 7 or? (Hint: find the LCD frctions) Adding, sutrcting frctions: if the denomintors re the sme comine the numertors: exmple: Prolems 9: Find the sum or difference (reduce if possile): If the denomintors re different, find equivlent frctions with common denomintors, then proceed s efore: exmple: exmple: Multiplying frctions: multiply the top numers, multiply the ottom numers, reduce if possile. exmple: ( ) Dividing frctions: mke compound frction, then multiply the top nd ottom (of the ig frction) y the LCD of oth: exmple: exmple: B. Decimls: Mening of plces: in.9, ech digit position hs vlue ten times the plce to its right. The prt to the left of the point is the whole numer prt. Right of the point, the plces hve vlues: tenths, hundredths, etc.,.9 ( 00)+ ( 0)+ ( ) So, + ( 0)+ ( 00)+ ( 9 000)..Which is lrger:.9 or.7? To dd or sutrct decimls, like plces must e comined (line up the points). exmple:... exmple: +.. exmple: 6.0 (.)
2 $. $.68 Multiplying decimls: exmple:... exmple:...06 exmple: (.0) (.) Dividing decimls: chnge the prolem to n equivlent whole numer prolem y multiplying oth y the sme power of ten. exmple:..0 Multiply oth y 00, to get 0 0 exmple:.0 Multiply oth y 000, get C. Positive integer exponents nd squre roots of perfect squres: Mening of exponents (powers): exmple: 8 exmple: 6 Prolems : Find the vlue: (.). (.) is nonnegtive rel numer if 0 mens, where 0. Thus 9 7, ecuse 7 9. Also, D. Frctiondeciml conversion: Frction to deciml: divide the top y the ottom. exmple:.7 exmple: exmple: Prolems : Write ech s deciml. If the deciml repets, show the repeting lock of digits: Nonrepeting decimls to frctions: red the numer s frction, write it s frction, reduce if possile: exmple:. four tenths 0 exmple:.76 three nd seventy six hundredths Prolems 68: Write s frction: E. Percents: Mening of percent: trnslte percent s hundredths : exmple: 8% mens 8 hundredths or.08 or 8 00 To chnge deciml to percent form, multiply y 00: move the point plces right nd write the % symol. exmple:.07 7.% exmple:. % Prolems 960: Write s percent: To chnge percent to deciml form, move the point plces left nd drop the % symol. exmple: 8.76%.0876 exmple: 67%.67 Prolems 66: Write s deciml: 6. 0% 6..0% To solve percent prolem which cn e written in this form: % of is c First identify,,c : Prolems 66: If ech sttement were written (with the sme mening) in the form of % of is c, identify,, nd c : 6. % of 0 is is 0% of out of is % Given nd, chnge % to deciml form nd multiply (since of cn e trnslted multiply ). Given c nd one of the others, divide c y the other (first chnge percent to deciml, or if the nswer is, write it s percent).
3 exmple: Wht is 9.% of $000? ( % of is c : 9.% of $000 is? ) 9.% $000 $70 (nswer) exmple: 6 prolems correct out of 80 is wht percent? ( % of is c :? % of 80 is 6) % (nswer) exmple: 60 people vote in n election, which is 60% of the registered voters. How mny re registered? ( % of is c : 60 % of? is 60); 60%.6; (nswer) 66. % of 9 is wht? 67. Wht percent of 70 is 6? 68. % of wht is 60? F. Estimtion nd pproximtion: Rounding to one significnt digit: exmple:.67 rounds to exmple:.09 rounds to.0 exmple: 80 rounds to either 800 or 900 Answers: , , 8. (ecuse 0 8 < 8 ) TOPIC : POLYNOMIALS $ not rel numer Prolems 697: Round to one significnt digit To estimte n nswer, it is often sufficient to round ech given numer to one significnt digit, then compute. exmple: Round nd compute: is the estimte Prolems 77: Select the est pproximtion of the nswer: (, 0, 00, 000, 0000) (.0,.,.,, 0, 0) (,, 98, 0, 00) 7. (.6890) (, 6, 60, 000, 000) % % c % A. Grouping to simplify polynomils: The distriutive property sys: ( + c) + c exmple: ( x y ) x y (, x, c y)
4 x + 7x ( + 7)x x exmple: ( x,, c 7) exmple: + 6x +x Prolems : Rewrite, using the distriutive property:. 6( x ). ( ). x x Commuttive nd ssocitive properties re lso used in regrouping: exmple: x + 7 x x x + 7 x + 7 exmple: x + + x 0 x exmple: x + y x + y x x + y + y x + y Prolems 9: Simplify:. x + x 7. x ++ x x x y + y 8x 9. x y + y x B. Evlution y sustitution: exmple: If x, then 7 x 7 () 7 exmple: If 7 nd, then ( 7) ( ) 9( ) 9 exmple: If x, then x x ( ) ( ) Prolems 09: Given x, y, nd z. Find the vlue: 0. x. x + y. z 6. x x. xz 7. ( x + z). y + z 8. x + z. y + z 9. x z C. Adding, sutrcting polynomils: Comine like terms: exmple: ( x + x +) ( x ) x + x + x + x + exmple: ( x ) + ( x + x ) x + x + x x + x exmple: ( x + x ) ( 6x x +) x + x 6x + x x + x Prolems 0: Simplify: x + 0. x + x. ( x )+ ( x). ( )+ ( + ). ( y y ) ( y y + ). ( 7 x) ( x 7). x ( x + x ) D. Monomil times polynomil: Use the distriutive property: exmple: ( x ) x + ( ) x + ( ) x exmple: ( x + ) x + exmple: x( x ) x + x 0. ( x ) 6. x 7 7. ( ). ( x ) ( ) 8. x( x + ). 8( + 7) 9. ( x )7 E. Multiplying polynomils: Use the distriutive property: ( + c) + c ( x ) is + c exmple: x + if: ( x +), x, nd c So ( + c) + c ( x +)x + ( x +) ( ) x + x 8x x 7x Short cut to multiply ove two inomils (see exmple ove): FOIL (do mentlly nd write the nswer) F: First times First: ( x) ( x) x O: multiply Outers : ( x) ( ) 8x I: multiply Inners : ( x) x L: Lst times Lst: ( ) Add, get x 7x exmple: ( x + ) ( x + ) x + x + 6 exmple: ( x ) ( x + ) x + x exmple: ( x ) ( x + ) x exmple: ( x ) x + exmple: ( x ) 9x x +6 exmple: ( x + ) ( ) x x + Prolems : Multiply:. ( x + ) 8. 6x( x). ( x ) 9. ( x ). ( x + ) ( x ) 0. ( x ) ( x + ) 6. ( x + ) ( x ). ( x ) ( x + ) 7. ( x ) ( x )
5 F. Specil products: These product ptterns (exmples of FOIL) should e rememered nd recognized: I. ( + ) ( ) II. ( + ) + + III. + Prolems : Mtch ech pttern with its exmple:. ( x ) 9x 6x +. ( x + ) x + 0x + c. ( x + 8) ( x 8) x 6. I:. II:. III: Prolems : Write the nswer using the pproprite product pttern:. ( +) ( ) 9. ( ) ( ) 0. ( x y). ( x +y) ( ). ( x + y) x y 6. y G. Fctoring: Monomil fctors: + c ( + c) exmple: x x x( x ) exmple: x y + 6xy xy( x + ) Difference of two squres: + ( ) ( x ) exmple: 9x x + Trinomil squre: exmple: x 6x + 9 ( x ) Trinomil: exmple: x x ( x ) ( x +) exmple: 6x 7x x + Prolems 67: Fctor: ( x ) x x x x. 8x 6. 8x + 8x + x 6. x 0x x +x + 7. xy +0x 6. 6x y 9x y 8. x x 66. x x 9. x x x 0x x y y x Answers:. 6x 8. x. +. x. 6. x + y 7. x x 9. x y x. x.. y y 0. x. x + 6. x x + x 9. x 7 0. x. x x + 6x + 9. x 6x + 9. x 9 6. x 9 7. x 6x x + 6x 9. x x + 0. x + x. x + x. c y y x xy + y. 6x + xy + 9y. 9x y ( x ) ( x +) ( x + ) ( x + ) ( + x) ( x ). x + 6. x 7. x y x 8. x 9. x 60. xy x y 6. x 6. x x 6. x x + 6. x + 6. x y y x 66. x 67. x
6 6 TOPIC : LINEAR EQUATIONS nd INEQUALITIES A. Solving one liner eqution in one vrile: Add or sutrct the sme vlue on ech side of the eqution, or multiply or divide ech side y the sme vlue, with the gol of plcing the vrile lone on one side. If there re one or more frctions, it my e desirle to eliminte them y multiplying oth sides y the common denomintor. If the eqution is proportion, you my wish to crossmultiply. Prolems : Solve:. x 9 7. x 6 x. 6x 8. x x +. x x x x. 0. 7x x +0. x 9. x + x 6. x x + To solve liner eqution for one vrile in terms of the other(s), do the sme s ove: exmple: Solve for F : C F 9 Multiply y 9 : 9 C F Add : 9 C + F Thus, F 9 C + exmple: Solve for : + 90 Sutrct : 90 exmple: Solve for x : x + c Sutrct : x c Divide y : x c Prolems 9: Solve for the indicted vrile in terms of the other(s): y x x y x + x. P + h 8. x + y 0 x. y x 9. y x 0 x y B. Solution of onevrile eqution reducile to liner eqution: Some equtions which do not pper to e liner cn e solved y using relted liner eqution: exmple: x+ x Multiply y x : x + x Solve: x x (Be sure to check nswer in the originl eqution.) exmple: x+ x+ Think of s nd crossmultiply: x + x + x x But x does not mke the originl eqution true (thus it does not check), so there is no solution. Prolems 0: Solve nd check: 0. x x x+ x. x x+. x x+8. x x+. x x exmple: x Since the solute vlue of oth nd is, x cn e either or. Write these two equtions nd solve ech: x or x x x x or x Prolems 60: Solve: 6. x 9. x 0 7. x 0. x + 8. x C. Solution of liner inequlities: Rules for inequlities: If >, then: + c > + c c > c c > c (if c > 0 ) c < c (if c < 0 ) c > c (if c > 0 ) c < c (if c < 0 ) If <, then: + c < + c c < c c < c (if c > 0 ) c > c (if c < 0 ) c < c (if c > 0 ) c > c (if c < 0 ) exmple: One vrile grph: solve nd grph on numer line: x 7 (This is n revition for { x : x 7}) Sutrct, get x 6 Divide y, x Grph: Prolems 8: Solve nd grph on numer line:. x >. x <
7 . x x > x. < x 7. x > +. x < x + x D. Solving pir of liner equtions in two vriles: The solution consists of n ordered pir, n infinite numer of ordered pirs, or no solution. Prolems 96: Solve for the common solution(s) y sustitution or liner comintions: 7 9. x + y 7. x y x y 8 x + y 0. x + y. x y x y x + y. x y 9. x + y x 8 x + y. x y 6. x y y x 6x 9 y Answers: ( P h). ( y +). 6. y ( y ) y 9. x y TOPIC : QUADRATIC EQUATIONS no solution 6., 7. no solution 8., 9. 0.,. x > 7. x < x 0. x > x > 6. x < 7. x > x  9. ( 9, ) 0. (, ). (8, ). (, 9). 8 ( 9, 9 ). (,0) 0. no solution 6. ny ordered pir of the form (, ) where is ny numer. One exmple: (, ). Infinitely mny solutions. A. x + x + c 0: A qudrtic eqution cn lwys e written so it looks like x + x + c 0 where,, nd c re rel numers nd is not zero. exmple: x x Add x: x + x Sutrct : 0 x + x or x + x 0 So,, c exmple: x Rewrite: x 0 (Think of x + 0x 0) So, 0, c Prolems : Write ech of the following in the form x + x + c 0 nd identify,, c :. x + x 0. x x. x 0. x x. 8x B. Fctoring: Monomil fctors: + c + c exmple: x x x x exmple: x y + 6xy xy x + Difference of two squres: + ( ) ( x ) exmple: 9x x +
8 Trinomil squre: exmple: x 6x + 9 ( x ) Trinomil: ( x +) ( x ) exmple: x x x exmple: 6x 7x x + Prolems 60: Fctor: x x x x 8. 8x 6. x + 8x + 8x 9. x 0x x +x + 0. xy +0x 8. 6x y 9x y. x x 9. x x. x x 6 0. x 0x +. x y y x C. Solving fctored qudrtic equtions: The following sttement is the centrl principle: If 0, then 0 or 0. First, identify nd in 0: exmple: ( x) ( x + ) 0 Compre this with 0 ( x); ( x + ) Prolems : Identify nd in ech of the following:. x( x ) 0. ( x ) ( x ) 0. ( x )x 0. 0 ( x ) ( x +) Then, ecuse 0 mens 0 or 0, we cn use the fctors to mke two liner equtions to solve: exmple: If x( x ) 0 then( x) 0 or x 0 nd so x 0 or x ; x. Thus, there re two solutions: 0 nd exmple: If ( x) ( x + ) 0 then x or ( x + ) 0 nd thus x or x. 0 0 exmple: If x + 7 then x x 7 x 7 (one solution) 8 Note: there must e zero on one side of the eqution to solve y the fctoring method. Prolems : Solve:. ( x +) ( x ) 0 9. ( x 6) ( x 6) 0 6. x( x + ) 0 0. ( x ) ( x )x. x( x + ) x 8.0 x + ( x ) 0 D. Solving qudrtic equtions y fctoring: Arrnge the eqution so zero is on one side (in the form x + x + c 0), fctor, set ech fctor equl to zero, nd solve the resulting liner equtions. exmple: Solve: 6x x Rewrite: 6x x 0 Fctor: x x So x 0 or x 0 0 Thus, x 0 or x exmple: 0 x x 0 ( x ) ( x + ) x 0 or x + 0 x or x Prolems : Solve y fctoring: ( x + ) ( x ) ( x ) 0. x x. x x 0 9. x +. x x 0. 6x x +. x( x + ) x 6x 6. x x. x x 7. x + x 6. x x 6 0 Another prolem form: if prolem is stted in this form: One of the solutions of x + x + c 0 is d, solve the eqution s ove, then verify the sttement. exmple: Prolem: One of the solutions of 0x x 0 is A. B. C. D. E. Solve 0x x 0 y fctoring: x( x ) 0 so x 0 or (x ) 0 thus x 0 or x. Since x is one solution, nswer C is correct.
9 . One of the solutions of x A. B. C. 0 D. E. ( x + ) 0 is 9. One solution of x x 0 is A. B. C. D. E. Answers: c. x + x 0 . x x x x x x Note: ll signs could e the opposite ( x ) ( x +) ( x + ) 8. x + 9. x 0. x y x. x. x. xy x y TOPIC : GRAPHING ( x + ) ( + x) ( x ). x. x x 6. x x + 7. x + 8. x y y x 9. x 0. x. x x. x x. x x. x x +., 6., , 8., , 0,. 0,. 0,. 0,., 0 6., 7., 8., 9., 0.,..,.,. B. B A. Grphing point on the numer line: Prolems 7: Select the letter of the point on the numer line with coordinte: A B C D E F G Prolems 80: Which letter est loctes the given numer: Prolems : Solve ech eqution nd grph the solution on the numer line: exmple: x + x. x 6 0. x + x. x x + B. Grphing liner inequlity (in one vrile) on the numer line: Rules for inequlities: If >, then: + c > + c c > c c > c (if c > 0) c < c (if c < 0) (if c > 0) > c c < c c (if c < 0) If <, then: + c < + c c < c c < c (if c > 0) c > c (if c < 0) (if c > 0) < c c > c c (if c < 0) exmple: One vrile grph: solve nd grph on numer line: x 7 (This is n revition for { x : x 7}) Sutrct, get x 6 Divide y , x Grph:
10 0 Prolems 0: Solve nd grph on numer line:. x > 8. x < 6. x < 9. x > x 6. x x >+ 7. < x exmple: x > nd x < The two numers  nd splits the numer line into three prts: x >, < x <, nd x <. Check ech prt to see if oth x > nd x < re true: prt x vlues x >? x <? oth true? x < < x < no yes yes yes no yes (solution) x > yes no no Thus the solution is < x < nd the grph is: exmple: x or x < ( or mens nd/or ) prt x vlues x? x <? x x < x > yes no no yes yes no t lest one true? yes (solution) yes (solution) no So x or x <; these cses re oth covered if x <. Thus the solution is x < nd the grph is: Prolems : Solve nd grph:. x < or x >. x 0 nd x >. x > nd x C. Grphing point in the coordinte plne: If two numer lines intersect t right ngles so tht: ) one is horizontl with positive to the right nd negtive to the left, ) the other is verticl with positive up nd negtive down, nd ) the zero points coincide Then they form coordinte plne, nd ) the horizontl numer line is clled the xxis, ) the verticl line is the yxis, ) the common zero point is the origin, ) there re four y qudrnts, II I numered x s shown: III IV To locte point on the plne, n ordered pir of numers is used, written in the form (x, y). The xcoordinte is lwys given first. Prolems 7: Identify x nd y in ech ordered pir:. (, 0) 6. (, ). (, ) 7. (0, ) To plot point, strt t the origin nd mke the moves, first in the xdirection (horizontl) nd the in the ydirection (verticl) indicted y the ordered pir. exmple: (, ) Strt t the origin, move left (since x ), then (from there), up (since y ) Put dot there to indicte the point (, ) 8. Join the following points in the given order: (, ), (, ), (, 0), (, ), (, ), (, 0), (, ), (, ), (, ) 9. Two of the lines you drew cross ech other. Wht re the coordintes of this crossing point? 0. In wht qudrnt does the point (, ) lie, if > 0 nd < 0? Prolems : For ech given point, which of its coordintes, x or y, is lrger? D. Grphing liner equtions on the coordinte plne: The grph of liner eqution is line, nd one wy to find the line is to join points of the line. Two points determine line, ut three re often plotted on grph to e sure they re colliner (ll in line). Cse I: If the eqution looks like x, then there is no restriction on y, so y cn e ny numer. Pick numers for vlues of y, nd mke ordered pirs so ech hs x. Plot nd join.
11 exmple: x Select three y s, sy , 0, nd Ordered pirs: (, ), (, 0), (, ) Plot nd join: Note the slope formul 0 gives ( ) 0, which is not defined: verticl line hs no slope. Cse II: If the eqution looks like y mx +, where either m or (or oth) cn e zero, select ny three numers for vlues of x, nd find the corresponding y vlues. Grph (plot) these ordered pirs nd join. exmple: y Select three x s, sy , 0, nd Since y must e , the pirs re: (, ), (0, ), (, ) The slope is ( ) And the line is horizontl. exmple: y x Select x s, sy 0,, : If x 0, y 0 If x, y If x, y Ordered pirs: (0, ), (, ), (, ) Note the slope is ( ) 0, And the line is neither horizontl nor verticl. Prolems : Grph ech line on the numer plne nd find its slope (refer to section E elow if necessry):. y x 9. x 6. x y 0. y x 7. x y 8. y. y x + E. Slope of line through two points: Prolems 7: Find the vlue of ech of the following: ( ) ( ) nd The line joining the points P x, y hs slope y y P x, y x x. exmple: A (, ), B (, ) slope of AB ( ) Prolems 8: Find the slope of the line joining the given points: 8. (, ) nd (, ) 9. (0, ) nd (, ) 0. (, ) nd (, ).. Answers:. D. E. C. F. B 6. G 7. B 8. Q 9. T 0. S... ½. x > 7. x < 6. x ½ 0 7. x > 6 8. x > 9. x < 0. x >. x < or x >. x >. < x x y (0,) 0. IV. x. y. y. x none  
12 0. . ½ . ½ none (undefined) TOPIC 6: RATIONAL EXPRESSIONS A. Simplifying frctionl expressions: exmple: (note tht you must e le to find common fctor in this cse 9 in oth the top nd ottom in order to reduce frction.) exmple: (common fctor: ) Prolems : Reduce: c x x ( x+) ( x ) ( x ) ( x ). 6xy y 0. x 9x x 9. 8( x ) 6 x x 7y x. x 7y x x+ exmple: y 0x 0 x y x y x y x x y y y y y Prolems : Simplify: x. xy y 6 y. x x x B. Evlution of frctions: exmple: If nd, find the vlue of + Sustitute: + () x( x ) x 6 Prolems : Find the vlue, given,, c 0, x, y, z : x y y x x c x 7.. z y 8. c. z C. Equivlent frctions: exmple: is equivlent to how mny eighths? 8, 6 8 exmple: exmple: x+ x+ ( x+) x+ x+ x+8 x+ x+ x+ exmple: x x+ ( x+) ( x ) x ( x ) ( x ) x+ ( x ) ( x+) x x+ ( x+) ( x ) Prolems 7: Complete: ( +) ( ). x 7 7y 7. x 6 6 x x+. x+ ( x ) ( x+) How to get the lowest common denomintor (LCD) y finding the lest common multiple (LCM) of ll denomintors: exmple: 6 nd 8. First find LCM of 6 nd : 6 LCM 0, so 6 0, nd exmple: nd 6 : 6 LCM, so 9, nd 6 exmple: x+ nd x LCM ( x + ) ( x ), so x+ x ( x+) ( x ), nd x x+ ( x+) ( x ) Prolems 8: Find equivlent frctions with the lowest common denomintor: 8. nd 9. x nd x 9. x nd. x nd x+ 0. x nd x+. x nd x x+
13 D. Adding nd sutrcting frctions: If denomintors re the sme, comine the numers: exmple: x x x x x y y y y Prolems 8: Find the sum or difference s indicted (reduce if possile): x+ x +x y xy. x x x If denomintors re different, find equivlent frctions with common denomintors, then proceed s efore (comine numertors): exmple: exmple: + x x+ ( x+) ( x ) ( x+) + ( x ) ( x ) ( x+) ( x+6+x x ) ( x+) x+ ( x ) ( x+) Prolems 9: Find the sum or difference: x 7. x + x x x. x 8. x x x x x x+ x. 0. x x x x. c. x x x. + E. Multiplying frctions: Multiply the tops, multiply the ottoms, reduce if possile: exmple: exmple: ( x+) x x x ( x+) ( x+) ( x ) ( x ) ( x+) ( x ) x+6 x ( ). c d ( x+). 9. x+ y y x 6 x x x+6 F. Dividing frctions: A nice wy to do this is to mke compound frction nd then multiply the top nd ottom (of the ig frction) y the LCD of oth: exmple: c d c d 7 exmple: d d c c d d x x exmple: x x y y x y y x y x x 9 x+7 x c c xy y Answers:... x.. 6. x y y c x ( x +) x. ( x ) ( x +). x+ x. x. x undefined xy ( x + ) ( ). x + x or x 6. + or , , x x x 0. x ( x + ) ( x +), ( x +)., x x
14 . ( x+) ( x ) ( x+), ( x ) ( x ) ( x +), x x( x +). x + x x x x x. x 0 x... c x +x 9. x ( x ) ( x+) ( x ) 0. x + x. x +x x.. c d y x 9. x x +7 x c 7. c TOPIC 7: EXPONENTS nd SQUARE ROOT A. Positive integer exponents: mens use s fctor times. ( is the exponent or power of.) exmple: mens, nd hs vlue. exmple: c c c c Prolems : Find the vlue:. 8. (.) ( ) exmple: Simplify: Prolems 8: Simplify:. x 7. x y B. Integer exponents: I. c +c II. c III. IV. V. c ( ) c c ( ) c c c c c c ( x) x VI. 0 (if 0) VII. Prolems 98: Find x: 9. x. 8 x 0. x. x x x. x 7. c. x 8. y y Prolems 9: Find the vlue: c x 9. 7x 0 x 6. c+ x c 0. x 7. 6x. 8. x x x x x. ( ). xy. x c+ x c C. Scientific nottion: exmple: if the zeros in the ten s nd one s plces re significnt. If the one s zero is not, write.80 0 ; if neither is significnt:.8 0. exmple:
15 exmple: 0 00 exmple: Note tht scientific form lwys looks like 0 n where <0, nd n is n integer power of 0. Prolems : Write in scientific nottion:. 9,000, Prolems 68: Write in stndrd nottion: To compute with numers written in scientific form, seprte the prts, compute, then recomine. exmple: (. 0 ) (.) exmple: exmple: Prolems 96: Write nswer in scientific nottion: D. Simplifiction of squre roots: (.9 0 )(. 0 7 ) 8. 0 if nd re oth nonnegtive ( 0 nd 0). exmple: 6 exmple: 7 exmple: If x 0, x 6 x Note: If x < 0, x 6 x mens (y definition) tht ), nd ) 0 Prolems 769: Simplify (ssume ll squre roots re rel numers): x x E. Adding nd sutrcting squre roots: exmple: + exmple: Prolems 707: Simplify: F. Multiplying squre roots: if 0 nd 0. exmple: 6 6 exmple: 6 exmple: ( ) ( ) 0 Prolems 779: Simplify: ( ) ( ) 79. Prolems 808: Find the vlue of x: x 8. x G. Dividing squre roots:, if 0 nd > 0. exmple: (or 8 ) Prolems 886: Simplify: If frction hs squre root on the ottom, it is sometimes desirle to find n equivlent frction with no root on the ottom. This is clled rtionlizing the denomintor. exmple: exmple: 9
16 Prolems 879: Simplify: Answers: x x 8. y y x c 6. x 6 7. x 8. x 9 9. x x 6. 8x y x x 67. x
17 7 TOPIC 8: GEOMETRY A. Formuls for perimeter P nd re A of rectngles, squres, prllelogrms, nd tringles: Rectngle with se nd ltitude (height) h: P + h A h h If wire is ent in this shpe, the perimeter P is the length of the wire, nd the re A is the numer of squre units enclosed y the wire. exmple: A rectngle with 7 nd h 8: P + h units A h squre units A squre is rectngle with ll sides equl, so the rectngle formuls pply (nd simplify). If the side length is s: P s A s s s exmple: A squre with side s cm hs P s! cm A s cm (sq. cm) A prllelogrm with se nd height h nd other side : A h h P + exmple: A prllelogrm hs sides nd 6; is the length of the ltitude perpendiculr to the side. P units A h 0 squre units In tringle with side lengths,, nd c, nd ltitude height h to side : P + + c c A h h h
18 exmple: P + + c units A h 0 (.8 ) squre units Prolems 8: Find P nd A:. Rectngle with sides nd 0.. Rectngle with sides. nd.. Squre with sides miles.. Squre with sides yrds.. Prllelogrm with sides 6 nd, nd height 0 (on side 6). 6. Prllelogrm, ll sides, ltitude Tringle with sides,, nd. Side is the ltitude on side. 8. Tringle shown: B. Formuls for circumference C nd re A of circle: A circle with rdius r (nd dimeter d r ) hs distnce round (circumference) C!d!r (If piece of wire is ent into circulr shpe, the circumference is the length of the wire.) exmple: A circle with rdius r 70 hs d r 0 nd exct circumference C!r! 70 0! units If! is pproximted y 7, C 0" #0( 7 ) # 0units (pprox.) If! is pproximted y., C! 0(.) units The re of circle is A!r exmple: If r 8, exct re is A!r! 8 6! squre units Prolems 9: Find the exct C nd A for circle with: 9. rdius r units 0. r 0 feet. dimeter d km Prolems : A circle hs re 9! :. Wht is its rdius length?. Wht is the dimeter?. Find its circumference. r 8 Prolems 6: A prllelogrm hs re 8 nd two sides ech of length :. How long is the ltitude to those sides? 6. How long re ech of the other two sides? 7. How mny times the P nd A of cm squre re the P nd A of squre with sides ll 6 cm? 8. A rectngle hs re nd one side 6. Find the perimeter. Prolems 90: A squre hs perimeter 0: 9. How long is ech side? 0. Wht is its re?. A tringle hs se nd height ech 7. Wht is its re? C. Pythgoren theorem: In ny tringle with 90 (right) ngle, the sum of the squres of the legs equls the squre of hypotenuse. (The legs re the two shorter sides; the hypotenuse is the longest side.) If the legs hve lengths nd, nd c is the hypotenuse length, then + c. In words: In right tringle, leg squred plus leg squred equls hypotenuse squred. exmple: A right tringle hs hypotenuse nd one leg. Find the other leg. Since leg + leg hyp, + x 9 + x x! 9 6 x 6 Prolems : Find the length of the third side of the right tringle:. one leg:, hypotenuse: 7. hypotenuse: 0, one leg: 8. legs: nd Prolems 6: Find x:
19 8. In right!rst with right ngle R, SR nd TS 6. Find RT. (Drw nd lel tringle to solve.) 9. Would tringle with sides 7,, nd e right tringle? Why or why not? Similr tringles re tringles which re the sme shpe. If two ngles of one tringle re equl respectively to two ngles of nother tringle, then the tringles re similr. exmple:!abc nd!fed re similr: The pirs of sides which correspond re AB nd FE, BC nd ED, AC nd FD. D A 6 C Prolems 0: Use this figure: B 0. Find nd nme two similr tringles.. Drw the tringles seprtely nd lel them.. List the three pirs of corresponding sides. If two tringles re similr, ny two corresponding sides hve the sme rtio (frction vlue): exmple: the rtio to x, or x, is the sme s y nd c z. Thus x y, c x z, nd c. Ech of y z these equtions is clled proportion.. Drw the similr tringles seprtely, lel them, 0 nd write 9 proportions for the corresponding sides. 8 Prolems 7: Solve for x: exmple: Find x y writing nd solving proportion: x, so cross multiply nd get x or x x 7. E AC 0; EC 7 BC ; DC x D A B C F 6 A E D A B C E x x x D. Grphing on the numer line: Prolems 8: Nme the point with given coordinte: A B C D E F G " ". "... Prolems 6: On the numer line ove, wht is the distnce etween the listed points? (Rememer tht distnce is lwys positive.) 6. D nd G 9. B nd C 7. A nd D 0. B nd E 8. A nd F. F nd G Prolems : On the numer line, find the distnce from:. 7 to. to 7. 7 to. to 7 Prolems 69: Drw sketch to help find the coordinte of the point : 6. Hlfwy etween points with coordintes nd. 7. Midwy etween nd. 8. Which is the midpoint of the segment joining 8 nd. 9. On the numer line the sme distnce from 6 s it is from 0. E. Coordinte plne grphing: To locte point on the plne, n ordered pir of numers is used, written in the form ( x, y). Prolems 606: Identify coordintes x nd y in ech ordered pir: 60. (,0) 6. (," ) 6. ", 6. ( 0,) To plot point, strt t the origin nd mke the moves, first in the xdirection (horizontl) nd 0
20 then the ydirection (verticl) indicted y the ordered pir. exmple: (, ) Strt t the origin, move left (since x!), then (from there), up (since y ), put dot there to indicte the point (, ) 6. On grph pper, join these points in order: (, ), (, ), (, 0), (, ), (, ), (, 0), (, ), (, ), (, ) Two of the lines drwn in prolem 6 cross ech other. Wht re the coordintes of the crossing point? 66. In wht qudrnt is the point (,) if > 0 nd < 0? Prolems 6769: ABCD is squre, with C(," ) nd D (","). Find: 67. the length A y B of ech side. 68. the coordintes of A. x 69. the coordintes of the midpoint of DC. D C Prolems 707: Given A( 0,), B(,0): 70. Sketch grph. Drw AB. Find its length. 7. Find the midpoint of AB nd lel it C. Find the coordintes of C. 7. Wht is the re of the tringle formed y A, B, nd the origin? Answers:. 0 units, 0 units (units mens squre units). units, 6 units. miles, 9 miles. yrds, 9 6 yrds. 0 un., 60 un un., 7 un un., 0 un. 8. un., 6 un. 9. 0" un., " un. 0. 0" ft., 00" ft.. " km, " km. 7.. ". 6. Cnnot tell 7. P is times, A is times No, ecuse 7 + " 0. "ABE ~ "ACD D. E A. AB, AC ; AE, AD ; BE, CD. 9. or. or D 9. E 0. C. F. B. G. B. F B A C x,y 0 6. x ",y 6. x,y " 6. x 0,y (0, ) 66. IV ", 69. (,") ( 6, ) 7. 0
1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationIdentify graphs of linear inequalities on a number line.
COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line.  When grphing firstdegree eqution, solve for the vrible. The grph of this solution will be single point
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationList all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.
Mth Anlysis CP WS 4.X Section 4.4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether  is root of 0. Show
More informationChapters Five Notes SN AA U1C5
Chpters Five Notes SN AA U1C5 Nme Period Section 5: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles
More informationChapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationMath 154B Elementary Algebra2 nd Half Spring 2015
Mth 154B Elementry Alger nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities RentHep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationAdvanced Algebra & Trigonometry Midterm Review Packet
Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.
More information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the xxis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More informationALevel Mathematics Transition Task (compulsory for all maths students and all further maths student)
ALevel Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length:  hours work (depending on prior knowledge) This trnsition tsk provides revision
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationNumber systems: the Real Number System
Numer systems: the Rel Numer System syllusref eferenceence Core topic: Rel nd complex numer systems In this ch chpter A Clssifiction of numers B Recurring decimls C Rel nd complex numers D Surds: suset
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More informationConsolidation Worksheet
Cmbridge Essentils Mthemtics Core 8 NConsolidtion Worksheet N Consolidtion Worksheet Work these out. 8 b 7 + 0 c 6 + 7 5 Use the number line to help. 2 Remember + 2 2 +2 2 2 + 2 Adding negtive number is
More informationShape and measurement
C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors  Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationSUMMER ASSIGNMENT FOR PreAP FUNCTIONS/TRIGONOMETRY Due Tuesday After Labor Day!
SUMMER ASSIGNMENT FOR PreAP FUNCTIONS/TRIGONOMETRY Due Tuesdy After Lor Dy! This summer ssignment is designed to prepre you for Functions/Trigonometry. Nothing on the summer ssignment is new. Everything
More informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
More informationAlgebra II Notes Unit Ten: Conic Sections
Syllus Ojective: 10.1 The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting
More informationStage 11 Prompt Sheet
Stge 11 rompt Sheet 11/1 Simplify surds is NOT surd ecuse it is exctly is surd ecuse the nswer is not exct surd is n irrtionl numer To simplify surds look for squre numer fctors 7 = = 11/ Mnipulte expressions
More information( ) Same as above but m = f x = f x  symmetric to yaxis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.
AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find
More informationLoudoun Valley High School Calculus Summertime Fun Packet
Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!
More informationMTH 416a Trigonometry
MTH 416 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on rightngled tringle. Identify the opposite, djcent nd hypotenuse in the following rightngled
More informationBEGINNING ALGEBRA (ALGEBRA I)
/0 BEGINNING ALGEBRA (ALGEBRA I) SAMPLE TEST PLACEMENT EXAMINATION Downlod the omplete Study Pket: http://www.glendle.edu/studypkets Students who hve tken yer of high shool lger or its equivlent with grdes
More informationCHAPTER 9. Rational Numbers, Real Numbers, and Algebra
CHAPTER 9 Rtionl Numbers, Rel Numbers, nd Algebr Problem. A mn s boyhood lsted 1 6 of his life, he then plyed soccer for 1 12 of his life, nd he mrried fter 1 8 more of his life. A dughter ws born 9 yers
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationVectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3dimensional vectors:
Vectors 1232018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2dimensionl vectors: (2, 3), ( )
More information7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?
7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More information8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.
8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationAlg. Sheet (1) Department : Math Form : 3 rd prep. Sheet
Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( 5, 9 ) ) (,
More informationDate Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )
UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationSummary Information and Formulae MTH109 College Algebra
Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)
More informationSuppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = 2.
Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot
More informationTO: Next Year s AP Calculus Students
TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC
More informationImproper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
More informationLesson 2.4 Exercises, pages
Lesson. Exercises, pges A. Expnd nd simplify. ) + b) ( ) () 0  ( ) () 0 c) 7 + d) (7) ( ) 7  + 8 () ( 8). Expnd nd simplify. ) b)  7  + 7 7( ) ( ) ( ) 7( 7) 8 (7) P DO NOT COPY.. Multiplying nd Dividing
More informationREVIEW SHEET FOR PRECALCULUS MIDTERM
. If A, nd B 8, REVIEW SHEET FOR PRECALCULUS MIDTERM. For the following figure, wht is the eqution of the line?, write n eqution of the line tht psses through these points.. Given the following lines,
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet  Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More informationQuotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m
Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble
More informationMathematics Number: Logarithms
plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationx means to use x as a factor five times, or x x x x x (2 c ) means to use 2c as a factor four times, or
14 DAY 1 CHAPTER FIVE Wht fscinting mthemtics is now on our gend? We will review the pst four chpters little bit ech dy becuse mthemtics builds. Ech concept is foundtion for nother ide. We will hve grph
More informationPrecalculus Spring 2017
Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify
More informationEquations and Inequalities
Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in
More informationTrigonometric Functions
Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds
More informationIntroduction to Algebra  Part 2
Alger Module A Introduction to Alger  Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger  Prt Sttement of Prerequisite
More informationFaith Scholarship Service Friendship
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
More informationThe graphs of Rational Functions
Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
More information0.1 THE REAL NUMBER LINE AND ORDER
6000_000.qd //0 :6 AM Pge 00 CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.
More informationR(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of
Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of
More informationSimplifying Algebra. Simplifying Algebra. Curriculum Ready.
Simplifying Alger Curriculum Redy www.mthletics.com This ooklet is ll out turning complex prolems into something simple. You will e le to do something like this! ( 9 # + 4 ' ) ' ( 9 + 7) ' ' Give this
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More information10.2 The Ellipse and the Hyperbola
CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More informationEdexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationOn the diagram below the displacement is represented by the directed line segment OA.
Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More information8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1
8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors PreChpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationSTRAND J: TRANSFORMATIONS, VECTORS and MATRICES
Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors
More informationUSA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year
1/1/21. Fill in the circles in the picture t right with the digits 18, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.
More informationI. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=
11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationSpecial Numbers, Factors and Multiples
Specil s, nd Student Book  Series H + 3 + 5 = 9 = 3 Mthletics Instnt Workooks Copyright Student Book  Series H Contents Topics Topic  Odd, even, prime nd composite numers Topic  Divisiility tests
More informationMATH STUDENT BOOK. 10th Grade Unit 5
MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY
More informationAlg 3 Ch 7.2, 8 1. C 2) If A = 30, and C = 45, a = 1 find b and c A
lg 3 h 7.2, 8 1 7.2 Right Tringle Trig ) Use of clcultor sin 10 = sin x =.4741 c ) rete right tringles π 1) If = nd = 25, find 6 c 2) If = 30, nd = 45, = 1 find nd c 3) If in right, with right ngle t,
More information3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression.
SECTION. Eponents nd Rdicls 7 B 7 7 7 7 7 7 7 NOW TRY EXERCISES 89 AND 9 7. EXERCISES CONCEPTS. () Using eponentil nottion, we cn write the product s. In the epression 3 4,the numer 3 is clled the, nd
More informationSummer Work Packet for MPH Math Classes
Summer Work Pcket for MPH Mth Clsses Students going into Preclculus AC Sept. 018 Nme: This pcket is designed to help students sty current with their mth skills. Ech mth clss expects certin level of number
More informationAP Calculus AB Summer Packet
AP Clculus AB Summer Pcket Nme: Welcome to AP Clculus AB! Congrtultions! You hve mde it to one of the most dvnced mth course in high school! It s quite n ccomplishment nd you should e proud of yourself
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More informationAT100  Introductory Algebra. Section 2.7: Inequalities. x a. x a. x < a
Section 2.7: Inequlities In this section, we will Determine if given vlue is solution to n inequlity Solve given inequlity or compound inequlity; give the solution in intervl nottion nd the solution 2.7
More informationA study of Pythagoras Theorem
CHAPTER 19 A study of Pythgors Theorem Reson is immortl, ll else mortl. Pythgors, Diogenes Lertius (Lives of Eminent Philosophers) Pythgors Theorem is proly the estknown mthemticl theorem. Even most nonmthemticins
More informationMDPT Practice Test 1 (Math Analysis)
MDPT Prctice Test (Mth Anlysis). Wht is the rdin mesure of n ngle whose degree mesure is 7? ) 5 π π 5 c) π 5 d) 5 5. In the figure to the right, AB is the dimeter of the circle with center O. If the length
More informationTriangles The following examples explore aspects of triangles:
Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x  4 +x xmple 3: ltitude of the
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
More informationPART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.
PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic
More informationI do slope intercept form With my shades on MartinGay, Developmental Mathematics
AATA Dte: 1//1 SWBAT simplify rdicls. Do Now: ACT Prep HW Requests: Pg 49 #1745 odds Continue Vocb sheet In Clss: Complete Skills Prctice WS HW: Complete Worksheets For Wednesdy stmped pges Bring stmped
More informationC Precalculus Review. C.1 Real Numbers and the Real Number Line. Real Numbers and the Real Number Line
C. Rel Numers nd the Rel Numer Line C C Preclculus Review C. Rel Numers nd the Rel Numer Line Represent nd clssif rel numers. Order rel numers nd use inequlities. Find the solute vlues of rel numers nd
More informationLesson 8.1 Graphing Parametric Equations
Lesson 8.1 Grphing Prmetric Equtions 1. rete tle for ech pir of prmetric equtions with the given vlues of t.. x t 5. x t 3 c. x t 1 y t 1 y t 3 y t t t {, 1, 0, 1, } t {4,, 0,, 4} t {4, 0,, 4, 8}. Find
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls 5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the lefthnd
More information5: The Definite Integral
5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce
More informationNat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS
Nt USAP This ooklet contins : Questions on Topics covered in RHS USAP Em Tpe Questions Answers Sourced from PEGASYS USAP EF. Reducing n lgeric epression to its simplest form / where nd re of the form (
More informationfractions Let s Learn to
5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin
More information