INTERMEDIATE ALGEBRA (ALGEBRA II) SAMPLE TEST PLACEMENT EXAMINATION

Transcription

1 9/0 INTERMEDIATE ALGEBRA (ALGEBRA II) SAMPLE TEST PLACEMENT EXAMINATION Downlod the omplete Study Pket: Students who hve tken yers of high shool lger or its equivlent with grdes of C or etter re eligile to tke this emintion. There re totl of questions on the emintion. The test is timed for minutes. No lultors re llowed. Smple questions from eh of the eight res elow re on the k of this sheet. Students who reeive stisftory sore my enroll in the following ourses: Mth / Finite Mthemtis Mth Lierl Arts Mthemtis Mth 8 Mth for Elementry Tehers Mth 00 College Alger Mth Clulus for Business Mth 6 Sttistis Mth 0 Pre-lulus The following topis re overed y the emintion:. Elementry Numeri & Algeri Opertions. Rtionl Epressions. Eponents nd Rdils. Liner Equtions; Inequlities; Asolute Vlue. Polynomils; Qudrti Equtions 6. The Co-ordinte Plne & Grphing 7. Funtions nd Logrithms 8. Word Prolems

2 Typil questions from eh of the ompeteny res of the Intermedite Alger Test. Elementry numeri lgeri opertions. + d (A) + d d (B) + d + (C) + d (D) + d (E). Rtionl Epressions d d (A) d d (B) d d (C) d (D) _ d (E) d.eponents nd Rdils + 7 (A) 6 (B) (C) (D) 0 (E) 0. Liner Equtions; inequlities; solute vlues. If + y 8 nd y _, then (A) _ 6 (B) 6 (C) 7 (D) 9 (E). Polynomils; qudrti equtions. One of the roots of ( ) ( + ) 0 is (A) _ (B) _ (C) _ (D) (E) 6. The oordinte plne nd grphing. Whih of the following is n eqution of line with slope nd y-interept? (A) y _ (B) y (C) y + (D) y _ (E) y + 7. Funtions nd logrithms. If log 0 + log 0 y, then y (A) 0.00 (B).0 (C) 0 (D) 00 (E), Word Prolems A student who orretly nswered 7 questions on test reeived sore of 7%. How mny questions were on the test? (A) (B) 7 (C) 7 (D) 96 (E) 0

3 Intermedite Alger Dignosti Test Prtie Topi : Elementry Opertions Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk 9. mth teher y + or y someone else who understnds this topi. A. Algeri Opertions, Grouping, Evlution: to evlute n epression, first do powers, then multiply nd divide in order from left to right, nd finlly dd nd sutrt in order from left to right. Prentheses hve preferene.: e d e d + Emple: ) 9 ) ) ) to 7: Find the vlue: ( ) to : If,, 0, d, nd e : 8. e 9. e + (d ) 0. ( d) + e... d e Comine like terms when possile: Emple: ) + y ( + y ) + y y y ) + to 0: Simplify: ( t) 6. 0r (r y) 7. ( ) (( ) ) 9. ( + ) ( ) ( 0 ) B. Simplifying Frtionl Epressions: Emple: ) (note tht you must e le to find ommon ftor in this se 9 in oth the top nd ottom in order to redue frtion.) ) (ommon ftor: ) to : Redue: ( + ) ( ) ( )( ) 6y y ).. 9 ( ) 6 7y 6.. 7y + to : Simplify: y y.. 6 y B. Evlution of Frtions + I. IV. ( ) II. Emple: y 0 0 y y y y y y y y V. ( ) 6 III. ( ) VI. 0 (if 0 ) to : Find : VII. 0 e ( ). y y Copyright 986, Ron Smith/Bishop Union High Shool, Bishop, CA 9 Permission grnted to opy for lssroom use only. Not to e sold One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 900.

4 Intermedite Alger Dignosti Test Prtie Topi : Elementry Opertions to 9: Simplify: E. Asolute vlue: Emple: 8 ) 6.. ) ) depends on ' ': ( + ) if 0, ; if < 0, ) 7 to 78: Find the vlue: 0. ( ) 8. ( ) ( ) (y ) ( y) to 8: If, find: C. Sientifi Nottion Emples: ) if the zeros in the ten s nd one s ples re signifint. If the one s zero is not, write.80 0, if neither is ( ( )) signifint:.8 0 Answers: ) ( + ) ) ) ( ).. ( + ) Note tht sientifi form lwys looks like 0 n + 9. / where < 0, nd n is n integer power of to 6: Write in sientifi nottion: 60. 9,000, to 66: Write in stndrd nottion: / To ompute with numers written in sientifi form, seprte the prts, ompute, then reomine. Emples: ) (. 0 )() (.)() ) ) to 7: Write nswer in sientifi nottion: ( 0 ) (. 0 ) (.9 0 )(. 0 ) / no vlue t y. y / if > 0 if < 0. / 9. (no vlue if 0). / / / y y /

5 Intermedite Dignosti Test Prtie Topi : Rtionl Epressions Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi. A. Adding nd Sutrting Frtions: if denomintors re the sme, omine the numertors: to : Find the sum or differene s indited (redue if possile):..... If denomintors re different, find equivlent frtions with ommon denomintors: to 0: Complete:? ) 6. + ( + )( ) to 0: Complete: Emple: + + y + y + y y y y y Emples: ) is the equivlent to how mny eighths? ) )? ? ? 7? 7 7y? ( + ) +? ? 6 ( )( ) + ( )( + ) ( )( + ) ( )( )? ( + )( ) How to get the lowest ommon denomintor (LCD) y finding the lest ommon multiple (LCM) of ll denomintors. + nd 6( + ) 6( + )( + ) to 6: Find equivlent frtions with the lowest ommon denomintor:. nd. nd. nd.. nd 6. After finding equivlent frtions with ommon denomintors, proeed s efore (omine numertors): Emples: ) ) Emples: ) 8 nd 6 : First find the LCM of 6 nd : LCM 0, so, nd ) nd : 6 6 LCM, so 9, nd 6 ) nd ( + ) 6( + ) ( + ) ( + ) so : ( ) ( )( ) ( )( ) ( ) ( + ) ( )( + ) (y ) nd ( ) 0( ) 7 to 0: Find the sum or differene ( ). ( )( ) , nd + ( ) ( )( + ) + ( )( + ) ( )( + ) + ( ) ( )( ) Copyright 986, Ron Smith/Bishop Union High Shool, Bishop, CA 9 Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 900.

6 Intermedite Dignosti Test Prtie Topi : Rtionl Epressions B. Multiplying Frtions: Multiply the tops, multiply the ottoms, redue if possile: Emples: ) ) ( + ) ( + )( + )( ) + ( )( + )( ) to 8: Multiply, redue if possile ( + ).. 8 ( y). 6. d ( + ) y 8. y 6 C. Dividing Frtions: A nie wy to do this is to mke ompound frtion nd then multiply the top nd ottom (of the ig frtion) y the LCD of oth: Emples: ) ) ) d 7 y d d d d d y y y y y y 9 to : Simplify: y (p ) p ( + ) Answers: / y 6. 9, /6,.. /8 ( + )., ( + ) ( + )., 9.. ( ) + 7,. 0( )( ). + ( y )( ) 0( )( ). +,, 6. ( + ) ( + ) 6. ( + ) 7. 8/ ( ) ( + )( ) d y ( + )( p) y / /8 0. / / 6

7 Intermedite Alger Dignosti Test Prtie Topi : Eponents nd Rdils Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi. + ( ) I. IV. II. A. Definitions of Powers nd Roots: to 0: Find the vlue..... ( ) B. Lws of Integer Eponents ( ) V. III. ( ) VI. 0 (if 0 ) VII. 7 7 to 0: Find :..... ( ) y y to : Find the vlue: ( ) ( ) to 7: Write the given two wys: no negtive no given powers frtions d d y yz y6z C. Lws of Rtionl Eponents, Rdils: Assume ll rdils re rel numers: I. If r is positive integer, p is n integer, nd 0, then II. III. IV. r p / r p ( r ) p whih is rel numer. (Also true if r is positive odd integer nd < 0.) p Think of s power. r root 8 to : Write s rdil: 8. / 9. / 0. (/) /. /. /. () / to 7: Write s frtionl power: r r r or ( ) / r / r / r r r r rs r s / rs, or, s r / r / r / r or r / s ( / s ) / ( / r ) Copyright 986, Ron Smith/Bishop Union High Shool, Bishop, CA 9 Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 900.

8 Intermedite Alger Dignosti Test Prtie Topi : Eponents nd Rdils 8 to 6: Find : E. Rtionliztion of Denomintors: Emples: ) ) / y + 7. if 0; / + ). 7 if < 0 6 to 6: Write with positive +. /8 7. eponents: ( 9 6 ) / y / ( 8 6 ) 8 to 9: Simplify: y D. Simplifition of Rdils: y Emples: y ) ( + y) 8. ) / 86. ) ). 8 dd dd 6 to 8: Simplify (ssume ll. y 9 6y rdils re rel numers): y9z 7. y9z Answers: / / / / / if 0; if < / 6. / y /8 y y ( + y) ( + y). 66.

9 Intermedite Alger Dignosti Test Prtie Topi : Liner Equtions nd Inequlities Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi. A. Solving One Liner Eqution in One Vrile: Add or sutrt the sme thing on eh side of the eqution, or multiply or divide eh side y the sme thing, with the gol of getting the vrile lone on one side. If there re one or more frtions, it my e desirle to eliminte them y multiplying oth sides y the ommon denomintor. If the eqution is proportion, you my wish to ross-multiply. to : Solve: To solve liner eqution for one vrile in terms of the other(s), do the sme s ove: Emples: ) Solve for F : C 9 (F ) 9 9 Multiply y : C F Add : Thus, F 9 C + 9 C + F ) Solve for : + 90 Sutrt : 90 6 to : Solve for the indited vrile in terms of the other(s): y y 8. P + h. y + B. Solving Pir of Liner Equtions in Two Vriles: the solution onsists of n ordered pir, n infinite numer of ordered pirs, or no solution.. y y 8. y 6. y 9. + y C. Anlyti Geometry of One Liner Eqution in Two Vriles: The grph of y m + is line with slope m nd y-interept. To drw the grph, find one point on it (suh s (0, ) ) nd then use the slope to find nother point. Drw the line joining the two. Emple: y + hs slope nd y-interept. To grph the line, lote (0, ). From tht point, go down (top of slope frtion), nd over (right) (ottom of frtion) to find seond point. Join. 0 to : Find slope nd y-interept, nd sketh the grph: 0. y +. y. y 8. y. y + A vertil line hs no slope, nd its eqution n e written so it looks like k (where k is numer). A horizontl line hs zero slope, nd its eqution looks like y k. Emple: Grph on the sme grph: nd + y. The first eqution is. The seond is y. to 6: Grph nd write equtions for:. the line thru (,) nd (, ). 6. the horizontl line thru (, ). y + y + y + y y 6 9 y to 9: Solve for the ommon solution(s) y sustitution or liner omintions:. + y 7. y 8 + y y Copyright 986, Ron Smith/Bishop Union High Shool, Bishop, CA 9 Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 900.

10 Intermedite Alger Dignosti Test Prtie Topi : Liner Equtions nd Inequlities D. Anlyti Geometry of Two Liner Equtions in Two F. Liner Inequlities: Vriles: Two distint lines in plne re either Rules for inequlities: prllel or interseting. They re prllel if nd If >, then: If <, then: only if they hve the sme slope, nd hene the + > + + < + equtions of the lines hve no ommon solutions. If the lines hve unequl slopes, they interset in one > < point nd their equtions hve etly one ommon > (if > 0) < (if > 0) solution. (They re perpendiulr iff their slopes re < (if < 0) > (if < 0) negtive reiprols, or one is horizontl nd the other is vertil.) If one eqution is multiple of > (if > 0) < (if > 0) the other, eh eqution hs the sme grph, nd every solution of one eqution is the solution of the < (if < 0) > (if < 0) other. 7 to : For eh pir of equtions in prolems to 9, tell whether the lines re prllel, perpendiulr, interseting ut not perpendiulr, or the sme line: 7. Prolem. Prolem E. Solution of One-Vrile Eqution Reduile to Liner Eqution: Some equtions whih don t pper liner n e solved y using relted liner eqution. to 9: Solve: to : Solve nd hek: 0.. Emple: Sine the solute vlue of oth nd is, n e either or. Write these two equtions nd solve eh: or or Emples: ) Squre oth sides: Solve: 6 Be sure to hek nswer(s): (hek) ) Squre: 9 Chek: There is no solution, sine 9 doesn t stisfy the originl eqution (it is flse tht 9 ). + Emple: One vrile grph: Solve nd grph on numer line: 7 (This is n revition for: {: 7 }) Sutrt, get 6 Divide y, Grph: to 9: Solve nd grph on numer line:. > 7. < 6. < 8. > > + 6. < 0 Answers:. m. 9/. /. /. m. /.. m / / 6. / / / 6. y. 6/. 6/ 7. int., not. / int., not. 0. int., not int., not int., not 8. F/ h. prllel 9. (y + )/. sme line 0. y. {, }. (y )/ 6. no solution. (9, ) 7. {, }. (, ) 8. {/}. (8, ) 9. {, }. (, 9) 0. { } 6. (8/9, /9). no solution 7. (/, 0). {/} 8. no solution. > (, ), where is ny numer;. < ½ 0 Infinite # of solutions. / 0 0. m 6. > m / 7. > 0 8. < 9. > 6 0

11 Intermedite Alger Dignosti Test Prtie Topi : Qudrti Polynomils, Equtions nd Inequlities Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi. E. Multiplying Polynomils Emples: ) ( +)( + ) ) ( )( + ) + ) ( )( + ) ) ( ) + ) ( + )( + ) + 8 6) ( ) ( )( ) ) ( + )( ) + to 0: Multiply.. ( + ). ( ). ( + )( ). ( +)( ). ( )( ) 6. 6( ) 7. ( )( + + ) 8. ( ) 9. ( )( + ) 0. ( )( + ) B. Ftoring Monomil Ftors: + ( + ) Emples: ) ( ) ) y + 6y y( + ) Differene of Two Squres: ( + )( ) Emple: 9 ( + )( ) Trinomil Squre: + + ( + ) + ( ) Emple: ( ) Trinomil: Emples: ) ( )( + ) ) 6 7 ( +)( ) Sum nd Differene of Two Cues: + ( + )( + ) ( )( + + ) Emple: 6 ( )( + + 6) to 7: Ftor ompletely: y y y y 9 y C. Solving Qudrti Equtions y Ftoring: If 0, then 0 or 0. Emple: If ( )( + ) 0 then ( ) 0 or ( + ) 0 nd thus or Note: there must e zero on one side of the eqution to solve y the ftoring method. Emple: 6 Rewrite: 6 0 Ftor: ( ) 0 So, 0 or ( ) 0 Thus, 0 or / 8 to 9: Solve y ftoring: 8. ( ) ( + ) ( + )( ) 0. ( + )( ) D. Completing the Squre: + will e the squre of inomil when is dded, if is found s follows: find hlf the oeffiient nd squre it -- this is. Thus, nd Emple: If the oeffiient of is not, ftor so it is. 0 to : Complete the squre, nd tell wht must e dded: E. The Qudrti Formul: if qudrti eqution looks like + + 0, then the roots (solutions) n e found y using the qudrti formul: to 9: Solve Hlf of is /, nd (/) /, whih must e dded to omplete the squre: Emple: Hlf of / is /6, nd ( /6) /6, so Thus, or must e dded to 6 to omplete the squre. ±

12 Intermedite Alger Dignosti Test Prtie Topi : Qudrti Polynomils, Equtions nd Inequlities F. Qudrti Inequlities Emple: Solve < 6. First mke one side zero: 6 < 0. Ftor: ( )( + ) < 0. If ( ) 0 or ( + ) 0, then or. These two numers ( nd ) split the rel numers < - < < into three sets (visulize the numer line): 0 ( ) ( + ) ( )( + ) solution? < negtive negtive positive no < < negtive positive negtive yes > positive positive positive no Therefore, if ( )( + ) < 0, then < <. Note tht this solution mens tht > nd < 0 to : Solve, nd grph on numer line: G. Comple Numers: is defined to e i, so i. Emple: i i - i i -i. Find the vlue of i > 0. <. + > 0. + < 0. > 66 to 67: Solve nd write the nswer s + i >. {, }. {, }. {, }. { /, /} 6. { /, /} 7. {} 8. {, /} 9. {, } 0. ( ), dd. ( + ½), dd ¼. 9, dd 6. ( + ), dd 8. {, }. { } 6. ± 7 7. {, } 8. ± 9. ± 0. < or > A omple numer is of the form + i, where nd re rel numers. is lled the rel prt nd is the imginry prt. If is zero, + i is rel numer. If 0, then + i is pure imginry. Comple numer opertions: Emples:. ( + i) + ( i) i. ( + i) ( i) + i. ( + i)( i) 6 7i i 6 7i + 9 7i + i + i. i i + i 6 to 6. Write eh of the following so the nswer is in the form + i. 6. ( + i)( i) 7. ( + i) + ( i) 8. ( + i) ( i) ( + i) ( i) 60. i 6. i 8 6. i 6 6. i i 7 6. i 6 + i i i i Answers: ¼ ( + ). ( + ). ( + )( ). ( ). ( y + ) 6. ( )( + ) 7. ( )( + ) 8. y( y) 9. ( + )( + ) 0. ( )( + ). ( ). ( + ). ( + ). y(y ). ( )( + ) 6. ( )( ) 7. ( + )( + )( ) 8. {0, } 9. {0, } 0. {0, /}. {, 0} < < no solution no grph. 0 < < < or > / i i i i 6. i 66. ± i i + ± 7 i

13 Intermedite Dignosti Test Prtie Topi 6: Grphing nd the Coordinte Plne Diretions: Study the emples, work the prolems, then hek your nswers 6 to : on the Solve k for of this the sheet. indited If you vrile don t get the in nswer terms given, hek your work nd look for mistkes. If you hve troule, sk mth of teher the other(s): or someone else who understnds this topi. A. Grphing Points:. Join the following points in the given order: (, ), (, ), (, 0), (, ), (, ), (, 0), (, ), (, ), (, ). In wht qudrnt does the point (, ) lie, if > 0 nd < 0? to 6: For eh given point, 6 whih of its oordintes, or y, is lrger? B. Distne etween points: the distne etween the points P (, y ) nd P (, y ) is found y using the Pythgoren Theorem, whih gives P P ( ) + ( y ). Emple: A(, ), B(, ) 7 to 0: Find the length of the segment joining the given points: 7. (, 0), (0, ) 8. (, ), (, ) 9. (, ), (0, ) 0. (, ), (, 6) C. Liner Equtions in Two Vriles, Slope, Interepts, nd Grphing: the line joining the points P (, y ) nd P (, y ) hs slope y y. to : Find the slope of the line joining the given points:. (, ) nd (, ).. (0, ) nd (, ) y AB ( ( ) ) + ( ) + ( ) 0 Emple: A(, ), B(, ) ( ) Slope of AB. (, ) nd (, ). To find the -interept (-is rossing) of n eqution, let y e zero nd solve for. For the y-interept, let e zero nd solve for y. Emple: y If 0, y so the y-interept is. If y 0, so the -interept is. The grph of y m + is line with slope m nd y-interept. To drw the grph, find one point on it [suh s (0, )] nd then use the slope to find nother point. Drw the line joining the two. Emple: y + hs slope nd y-interept. To grph the line, lote (0, ). From tht point, go down (top of slope frtion), nd over (right) (ottom of frtion) to find seond point. Join. 6 to 0: Find slope nd y-interept, nd sketh the grph: 6. y + 9. y 7. y 0. y + 8. y 8 To find n eqution of non-vertil line, it is neessry to know its slope nd one of its points. Write the slope of the line thru (, y) nd the known point, then write n eqution whih sys tht this slope equls the known slope. Emple: Find n eqution of the line thru (, ) nd (, 0). 0 Slope + y 0 Using (, 0) nd (, y), slope ; + ross-multiply, get y +, or y to : Find n eqution of the line:. thru (, ) nd (, ). thru (0, ) nd (, ). thru (, ) nd (, ). thru (8, 0), with slope. thru (0, ), with slope / Copyright 986, Ron Smith/Bishop Union High Shool, Bishop, CA 9 Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted B. Solving with topis Pir on dignosti of Liner tests Equtions supplied in to the Two Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 900.

14 Intermedite Dignosti Test Prtie Topi 6: Grphing nd the Coordinte Plne A vertil line hs no slope, nd its eqution n e to 0: Sketh the grph: written so it looks like k (where k is numer). A horizontl line hs zero slope, nd its eqution looks like y k... y y 8. y ( + ) 9. y ( ) Emple: Grph on the sme grph: nd + y. The first eqution is. 6. y + 0. y ( + )( ) 7. y Answers:.. The seond is y. 6 to 7: Grph nd write equtions for: 6. the line thru (,) nd (, ). 7. the horizontl line thru (, ). D. Liner Inequlities in Two Vriles: Emple: Two vrile grph: grph solution on numer plne: y >. (This is n revition for {(,y): y > }. Sutrt, multiply y, get y <. Grph y, ut drw dotted line, nd shde the side where y <.. IV... y. y /. 7/.. 0. /. / 6. m m / 8 to : Grph: 8. y <. < y + 9. y >. + y < 0. y +. y > E. Grphing Qudrti Equtions: The grph of y + + is prol, opening upwrd (if > 0) or downwrd (if < 0), nd with line of symmetry, lso lled is of symmetry. To find the verte V(h, k) of the prol, h (sine V is on the is of symmetry), nd k is the vlue of y when h is sustituted for. Emples: ) y 6, 6, 0 6 Ais: h, k 8 9 Thus, verte is (, 9) 8. m m 0. m / / / 7.. y / / 8.. y. y. y + 8. y / y (, ) ) y ; V(0, ), Ais: 0 y-interept: if 0, y 0 -interept: if y 0, 0 so,, And ± 0.

15 Intermedite Alger Dignosti Test Prtie Topi 7: Logrithms nd Funtions Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi. A. Funtions: The re A of squre depends on its side length s, nd we sy A is funtion of s, nd write A f(s) ; for short, we red this A f of s. There re mny funtions of s. The one here is s. We write this f(s) s nd n trnslte: the funtion of s we re tlking out is s. Sometimes we write A(s) s. This sys the re A is funtion of s, nd speifilly, it is s. B. Funtion Vlues nd Sustitution: If A(s) s, A(), red A of, mens reple every s in A(s) s with, nd find A when s is. When we do this, we find A() 9. Emples: g() is given: y g() π ) g() π 9π ) g(7) π 7 9π ) g() π ) g( + h) π ( + h) π + π h + π. Given y f(). Complete these ordered pirs: (, ), (0, ), (/, ), (, 0), (, ), (, ) to 0: Given f() +. Find:. f(0) 7. f(). f() 8. f( ). f( ) 9. f(). f( ) 0. f() 6. f() to : Given f(). Find: +. f(). f( ). f( ). f(0). f( ) 6 to 9: Find ll rel zeros of: h Emple: If k(), for wht is k() 0? If k() 0, then 0 nd sine ( ) 0, n e either 0 or. (These vlues of : 0 nd, re lled zeros of the funtion, euse eh mkes the funtion zero.) 6. ( + ) to : Given: f() +, find rel so tht: 0. f(). f(). f(). is zero of f() Sine y f(), the vlues of y re the vlues of the funtion whih orrespond to speifi vlues of. The heights of the grph ove (or elow) the -is re the vlues of y nd so lso of the funtion. Thus for this grph, f() is the height (vlue) of the funtion t nd the vlue is : At, the vlue (height) of f() is zero; in other words, f( ) 0. Note tht f() > f( ), sine > 0, nd tht f(0) < f( ), sine f( ) nd f(0) <. to 8: For this grph, tell whether the sttement is true or flse:. g( ) g(0). g(0) g() 6. g() > g( ) 7. g( ) > g() 8. g() < g(0) <g() C. Logrithms nd Eponents: Eponentil form: 8 Logrithmi form: log 8 Both of the equtions ove sy the sme thing. log 8 is red log se two of eight equls three nd trnsltes the power of whih gives 8 is. 9 to : Write the following informtion in oth eponentil nd logrithmi forms: 9. The power of whih gives 9 is. 0. The power of whih gives is.. 0 to the power is. 00. is the power of 69 whih gives. to 8: Write in logrithmi form: y Copyright 986, Ron Smith/Bishop Union High Shool, Bishop, CA 9 Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 900.

16 Intermedite Alger Dignosti Test Prtie Topi 7: Logrithms nd Funtions 9 to : Write in eponentil form: 8. log ( 6) log (6 ) 9. log 9. log log 8. log log 0. log 0. y log 8. log (7 ). log. log log log to 0: Find the vlue: log log 0 log log 0 9. log log log log log D. Logrithm nd Eponent Rules: 88. log Eponent Rules: Log Rules: (se ny 9. (ll quntities rel) positive rel # eept ) Answers: + log log + log. 7,, /,,. y + z /,. z y log log log.. y / y. 6. log y ( ) log log y / y y /z ( ) log y yz ( log ) z log log y, (if 0 ) log z/y (se hnge rule). / 6. y/. 6. / p / r r p p power. no vlue 66. / p ( r ) think of s r root. 67. to : Given log 0 0, find:. log 0. log to 6: Solve for in terms of y nd z :. y z 7. log log y. 9 y z 6. log log y log z. y 8. log log y 6. y 60. log log y + log z 6. log + log y log z 6. log 7 y ; log 7 z ; log 6. y log 9 ; log E. Logrithmi nd Eponentil Equtions: 6 to 9: Use the eponent nd log rules to find the vlue of : log log log log 6 7. log (log ) log 0 log 0 + log log log 8 + log log 79. log 80. log log 0 ( ) 6., , / 69. / none 7. / , none 7.. ± P T T T T ; log ; log 8. ny rel numer. 0 /00; > 0 nd log 0 / / ; 8. / log 69 / 8. /. log log log log 0 /0 89. / 7. log log y y 9.. log log. 0 (ny se; 6. 0 if se, 7. log ) / log y log log

17 Intermedite Alger Dignosti Test Prtie Topi 8: Word Prolems Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi.. / of /6 of / of numer is. Wht is the numer?. On the numer line, points P nd Q re units prt. Q hs oordinte. Wht re the possile oordintes of P?. Wht is the numer, whih when multiplied y, gives 6?. If you squre ertin numer, you get 9. Wht is the numer?. Wht is the power of 6 tht gives you 6 /? 6. Point X is on eh of two given interseting lines. How mny suh points X re there? 7. Point Y is on eh of two given irles. How mny suh points Y re there? 8. Point Z is on eh of given irle nd given ellipse. How mny suh Z re there? 9. Point R is on the oordinte plne so its distne from given point A is less thn. Show in sketh where R ould e. 0 to : 0. If the length of hord AB is nd the length of CB is C B 6, wht is AC? A O. If AC y nd CB z, How long is AB (in terms of y nd z )?. This squre is ut into two smller squres nd two non-squre retngles s shown. Before eing ut, the lrge squre hd re ( + ). The two smller squres hve res nd. Find the totl re of the two non-squre retngles. Do the res of the prts dd up to the re of the originl squre?. Find nd y : 0 y. In order to onstrut n equilterl tringle with n re whih is 00 times the re of given equilterl tringle, how long side should e used? to 6: nd y re numers, nd two s equls three y s.. Whih of or y must e lrger? 6. Wht is the rtio of to y? 7 to : A plne hs ertin speed in still ir. In still ir, it goes 0 miles in hours. 7. Wht is its (still ir) speed? 8. How long does it tke to fly 000 mi.? 9. How fr does the plne go in hours? 0. If the plne flies ginst 0 mph hedwind, wht is its ground speed?. If it hs fuel for 7. hours of flying time, how fr n it go ginst this hedwind? to : Georgie nd Porgie ke pies. Georgie n omplete 0 pies n hour.. How mny n he mke in one minute?. How mny n he mke in 0 minutes?. How mny n he mke in minutes?. How long does he tke to mke 00 pies? 6 to 8: Porgie n finish pies n hour. 6. How mny n she mke in one minute? 7. How mny n she mke in 0 minutes? 8. How mny n she mke in minutes? 9 to : If they work together, how mny pies n they produe in: 9. minute. 80 minutes 0. minutes. hours to : A nurse needs to mi some lohol solutions, given s perent y weight of lohol in wter. Thus in % solution, % of the weight would e lohol. She mies gm of % solution, y gm of 0% solution, nd 0 gm of pure wter to get totl of 0 gm of solution whih is 8% lohol.. In terms of, how mny gm of lohol re in the % solution?. The y gm of 0% solution would inlude how mny gm of lohol?. How mny gm of solution re in the finl mi (the 8% solution)? 6. Write n epression in terms of nd y for the totl numer of gm in the 8% solution ontriuted y the three ingredients (the %, 0% nd wter). 7. Use your lst two nswers to write totl grms eqution. 8. How mny gm of lohol re in the 8%? 9. Write n epression in terms of nd y for the totl numer of gm of lohol in the finl solution. 0. Use the lst two nswers to write totl grms of lohol eqution.. How mny gm of eh solution re needed? Copyright 986, Ron Smith/Bishop Union High Shool, Bishop, CA 9 Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 900.

18 Intermedite Alger Dignosti Test Prtie Topi 8: Word Prolems. Hlf the squre of numer is 8. Wht is the numer?. If the squre of twie numer is 8, wht is the numer?. Given positive numer. The squre of positive numer y is t lest four times. How smll n y e?. Twie the squre of hlf of numer is. Wht is the numer? 6 to 8: Hlf of is the sme s one-third of y. 6. Whih of nd y is the lrger? 7. Write the rtio :y s the rtio of two integers. 8. How mny s equl 0 y s? 9 to 0: A gthering hs twie s mny women s men. W is the numer of women nd M is the numer of men. 9. Whih is orret: M W or M W? 0. If there re women, how mny men re there? to : If A is inresed y %, it equls B.. Whih is lrger, B or the originl A?. B is wht perent of A?. A is wht perent of B? to 6: If C is deresed y 0%, it equls D.. Whih is lrger, D or the originl C?. C is wht perent of D?. 6. D is wht perent of C? 7 to 8: The length of retngle is inresed y % nd its width is deresed y 0%. 7. Its new re is wht perent of its old re? 8. By wht perent hs the old re inresed or deresed? 9 to 6: Your wge is inresed y 0%, then the new mount is ut y 0% (of the new mount). 9. Will this result in wge whih is higher thn, lower thn, or the sme s the originl wge? 60. Wht perent of the originl wge is this finl wge? 6. If the ove steps were reversed (0% ut followed y 0% inrese), the finl wge would e wht perent of the originl wge? 6. Find % of is wht perent of 88? 6. Wht perent of is 88? 6. is % of wht numer? 66. The 00 people who vote in n eletion re 0% of the people registered to vote. How mny re registered? 67. If you get 6 on 0-question test, wht perent is this? 68. Wht is the verge of 87, 6, 8, 9, nd 9? 69. If two test sores re 8 nd 60, wht minimum sore on the net test would e needed for n overll verge of 80? 70. The verge height of 9 people is 68 inhes. Wht is the new verge height if 78-inh person joins the group? 7 to 7: s vries diretly s P, nd P 6 when s. 7. Find s when P. 7. Find P when s. 7 to 7: A is proportionl to r, nd when r 0, A 00π. 7. Find A when r. 7. Find r when A 6π. 7. If is inversely proportionl to h, nd 6 when h, find h when. 76. If y, write the rtio :y s the rtio of two integers. 77. The length of retngle is twie the width. If oth dimensions re inresed y m, the resulting retngle hs 8 m. Wht ws the originl width? 78. After retngulr piee of knitted fri shrinks in length one m nd strethes in width m, it is squre. If the originl re ws 0 m, wht is the squre re? Answers: y. +, y /7. 9 y 70/7. /. 6, ,, or. 8. 0,,,, or. 9. Inside the irle 6. y A of rdius 7. / entered on A M W. y , yes: ( + ) %. 0/7. 80% y 6/. C. 0 times the. 66 / % originl side 6. 60%. 7. 7% 6. / 8. % derese 7. 0 mph 9. lower 8. /9 hrs % 9. 0 miles 6. sme (96%) mph miles 6. 6.%. / 6. 60% / min % 6. / / in. 9. / / π y 76. : y y