1. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES)

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1 Numbers nd Opertions, Algebr, nd Functions 45. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES) In sequence of terms involving eponentil growth, which the testing service lso clls geometric sequence, there is constnt rtio between consecutive terms. In other words, ech successive term is the sme multiple of the preceding one. For emple, in the sequence, 4, 8, 6,,..., notice tht you multipy ech term by to obtin the net term, nd so the constnt rtio (multiple) is. To solve problems involving geometric sequence, you cn pply the following stndrd eqution: r (n ) = T In this eqution: The vrible is the vlue of the first term in the sequence The vrible r is the constnt rtio (multiple) The vrible n is the position number of ny prticulr term in the sequence The vrible T is the vlue of term n If you know the vlues of ny three of the four vribles in this stndrd eqution, then you cn solve for the fourth one. (On the SAT, geometric sequence problems generlly sk for the vlue of either or T.) Emple (solving for T when nd r re given): The first term of geometric sequence is, nd the constnt multiple is. Find the second, third, nd fourth terms. nd term (T) = ( ) = = 6 rd term (T) = ( ) = = 9 = 8 4th term (T) = (4 ) = = 7 = 54 To solve for T when nd r re given, s n lterntive to pplying the stndrd eqution, you cn multiply by r (n ) times. Given = nd r = : nd term (T) = = 6 rd term (T) = = 6 = 8 4th term (T) = = 6 = 8 = 54 NOTE: Using the lterntive method, you my wish to use your clcultor to find T if nd/or r re lrge numbers. Emple (solving for when r nd T re given): The fifth term of geometric sequence is 768, nd the constnt multiple is 4. Find the st term (). ( 5) 4 = = = 768 = =

2 46 Chpter 5 Emple (solving for T when nd nother term in the sequence re given): To find prticulr term (T) in geometric sequence when the first term nd nother term re given, first determine the constnt rtio (r), nd then solve for T. For emple, ssume tht the first nd sith terms of geometric sequence re nd 048, respectively. To find the vlue of the fourth term, first pply the stndrd eqution to determine r : ( 6) r = 048 Eercise 5 r = 048 r r 5 5 = 048 = 04 r = 5 04 r = 4 The constnt rtio is 4. Net, in the stndrd eqution, let =, r = 4, nd n = 4, nd then solve for T : ( 4) 4 = T 4 = T 64= T 8 = T The fourth term in the sequence is 8. Work out ech problem. For questions, circle the letter tht ppers before your nswer. Questions 4 nd 5 re grid-in questions.. On Jnury, 950, frmer bought certin prcel of lnd for $,500. Since then, the lnd hs doubled in vlue every yers. At this rte, wht will the vlue of the lnd be on Jnury, 00? $7,500 $9,000 $6,000 $4,000 $48,000. A certin type of cncer cell divides into two cells every four seconds. How mny cells re observble seconds fter observing totl of four cells?,04,048 4,096 5,5 8,9. The seventh term of geometric sequence with constnt rtio is 448. Wht is the first term of the sequence? Three yers fter n rt collector purchses certin pinting, the vlue of the pinting is $,700. If the pinting incresed in vlue by n verge of 50 percent per yer over the three yer period, how much did the collector py for the pinting, in dollrs? 5. Wht is the second term in geometric series with first term nd third term 47?

3 Numbers nd Opertions, Algebr, nd Functions 47. SETS (UNION, INTERSECTION, ELEMENTS) A set is simply collection of elements; elements in set re lso referred to s the members of the set. An SAT problem involving sets might sk you to recognize either the union or the intersection of two (or more) sets of numbers. The union of two sets is the set of ll members of either or both sets. For emple, the union of the set of ll negtive integers nd the set of ll non-negtive integers is the set of ll integers. The intersection of two sets is the set of ll common members in other words, members of both sets. For emple, the intersection of the set of integers less thn nd the set of integers greter thn 4 but less thn 5 is the following set of si consecutive integers: {5,6,7,8,9,0}. On the new SAT, problem involving either the union or intersection of sets might pply ny of the following concepts: the rel number line, integers, multiples, fctors (including prime fctors), divisibility, or counting. Emple: Set A is the set of ll positive multiples of, nd set B is the set of ll positive multiples of 6. Wht is the union nd intersection of the two sets? The union of sets A nd B is the set of ll postitive multiples of. The intersection of sets A nd B is the set of ll postitive multiples of 6.

4 48 Chpter 5 Eercise Work out ech problem. Note tht question is grid-in question. For ll other questions, circle the letter tht ppers before your nswer.. Which of the following describes the union of the set of integers less thn 0 nd the set of integers greter thn 0? Integers 0 through 0 All integers greter thn 0 but less thn 0 All integers less thn 0 nd ll integers greter thn 0 No integers All integers. Set A consists of the positive fctors of 4, nd set B consists of the positive fctors of 8. The intersection of sets A nd B is set contining how mny members? 4. The set of ll multiples of 0 could be the intersection of which of the following pirs of sets? The set of ll multiples of 5 ; the set of ll multiples of The set of ll multiples of ; the set of 5 ll multiples of 5 The set of ll multiples of ; the set of ll multiples of 0 The set of ll multiples of ; the set of 4 ll multiples of The set of ll multiples of 5 ; the set of ll multiples of 4 5. For ll rel numbers, sets P, Q, nd R re defined s follows: P:{ 0} Q:{ 0} R:{ 0}. The union of sets X nd Y is set tht contins ectly two members. Which of the following pirs of sets could be sets X nd Y? The prime fctors of 5; the prime fctors of 0 The prime fctors of 4; the prime fctors of 5 The prime fctors of 9; the prime fctors of 8 The prime fctors of ; the prime fctors of 5 The prime fctors of 9; the prime fctors of 5 Which of the following indictes the intersection of sets P, Q, nd R? = ny rel number 0 0 = 0 0 0

5 Numbers nd Opertions, Algebr, nd Functions 49. ABSOLUTE VALUE The bsolute vlue of rel number refers to the number s distnce from zero (the origin) on the rel-number line. The bsolute vlue of is indicted s. The bsolute vlue of negtive number lwys hs positive vlue. Emple: = 0 4 The correct nswer is. = 5 = 5, nd = =. Performing subtrction: 5 = 4. The concept of bsolute vlue cn be incorported into mny different types of problems on the new SAT, including those involving lgebric epressions, equtions, nd inequlities, s well s problems involving functionl nottion nd the grphs of functions. Eercise Work out ech problem. Circle the letter tht ppers before your nswer = For ll integers nd b, where b 0, subtrcting b from must result in positive integer if: b is positive integer ( b ) is positive integer (b ) is negtive integer ( + b) is positive integer (b) is positive integer. Wht is the complete solution set for the inequlity > 4? > > 7 < < 7 < 7, > 7 <, > 7 4. The figure below shows the grph of certin eqution in the y-plne. Which of the following could be the eqution? = y y = y = y = + = y ( ) = 5. If f() =, then f 0

6 50 Chpter 5 4. EXPONENTS (POWERS) An eponent, or power, refers to the number of times tht number (referred to s the bse number) is multiplied by itself, plus. In the number, the bse number is nd the eponent is. To clculte the vlue of, you 4 ( ), the bse number is 4 ( ), you multiply 4 by itself three times: multiply by itself twice: = = 8. In the number nd the eponent is 4. To 6 clculte the vlue of ( ) = =. 8 An SAT problem might require you to combine two or more terms tht contin eponents. Whether you cn you combine bse numbers using ddition, subtrction, multipliction, or division before pplying eponents to the numbers depends on which opertion you re performing. When you dd or subtrct terms, you cnnot combine bse numbers or eponents: + b ( + b) b ( b) Emple: If =, then 5 = The correct nswer is. You cnnot combine eponents here, even though the bse number is the sme in ll three terms. Insted, you need to pply ech eponent, in turn, to the bse number, then subtrct: 5 = ( ) 5 ( ) ( ) = 4 + = 4 There re two rules you need to know for combining eponents by multipliction or division. First, you cn combine bse numbers first, but only if the eponents re the sme: b = (b) b = b Second, you cn combine eponents first, but only if the bse numbers re the sme. When multiplying these terms, dd the eponents. When dividing them, subtrct the denomintor eponent from the numertor eponent: y = ( + y) y = ( y) When the sme bse number (or term) ppers in both the numertor nd denomintor of frction, you cn

7 Numbers nd Opertions, Algebr, nd Functions 5 fctor out, or cncel, the number of powers common to both. Emple: Which of the following is simplified version of y y y y y 5 y 5 The correct nswer is. The simplest pproch to this problem is to cncel, or fctor out, nd y from numertor nd denomintor. This leves you with in the denomintor nd y in the denomintor. You should lso know how to rise eponentil numbers to powers, s well s how to rise bse numbers to negtive nd frctionl eponents. To rise n eponentil number to power, multiply eponents together: ( ) = y y Rising bse number to negtive eponent is equivlent to divided by the bse number rised to the eponent s bsolute vlue: =? To rise bse number to frctionl eponent, follow this formul: y = y Also keep in mind tht ny number other thn 0 (zero) rised to the power of 0 (zero) equls : Emple: 0 = [ 0] ( ) 4 = 6 8 The correct nswer is. ( ) 4 = ()() = = = 6 4

8 5 Chpter 5 Eercise 4 Work out ech problem. For questions 4, circle the letter tht ppers before your nswer. Question 5 is gridin question.. b c = bc bc b b c b. 4 n + 4 n + 4 n + 4 n = 4 4n 6 n 4 (n n n n) 4 (n+) 6 4n. Which of the following epressions is simplified form of ( ) 4? If =, then = 0 5. Wht integer is equl to 4 + 4?

9 Numbers nd Opertions, Algebr, nd Functions 5 5. FUNCTION NOTATION In function (or functionl reltionship), the vlue of one vrible depends upon the vlue of, or is function of, nother vrible. In mthemtics, the reltionship cn be epressed in vrious forms. The new SAT uses the form y = f() where y is function of. (Specific vribles used my differ.) To find the vlue of the function for ny vlue, substitute the -vlue for wherever it ppers in the function. Emple: If f() = 6, then wht is the vlue of f(7)? The correct nswer is 8. First, you cn combine 6, which equls 4. Then substitute (7) for in the function: 4(7) = 8. Thus, f(7) = 8. A problem on the new SAT my sk you to find the vlue of function for either number vlue (such s 7, in which cse the correct nswer will lso be number vlue) or for vrible epression (such s 7, in which cse the correct nswer will lso contin the vrible ). A more comple function problem might require you to pply two different functions or to pply the sme function twice, s in the net emple. Emple: If f() =, then f f = The correct nswer is. Apply the function to ech of the two -vlues (in the first instnce, you ll obtin numericl vlue, while in the second instnce you ll obtin n vrible epression: f = ( ) = = 4= 8 f = 4 = = Then, combine the two results ccording to the opertion specified in the question: f f 8 6 = =

10 54 Chpter 5 Eercise 5 Work out ech problem. Circle the letter tht ppers before your nswer.. If f() =, then for which of the following vlues of does f() =? If f() =, then f ( ) =. If f() = + 4, then f( + ) = If f() = nd g() = +, then g(f()) = If f() =, then f( ) ( f( ) ) =

11 Numbers nd Opertions, Algebr, nd Functions FUNCTIONS DOMAIN AND RANGE A function consists of rule long with two sets clled the domin nd the rnge. The domin of function f() is the set of ll vlues of on which the function f() is defined, while the rnge of f() is the set of ll vlues tht result by pplying the rule to ll vlues in the domin. By definition, function must ssign ectly one member of the rnge to ech member of the domin, nd must ssign t lest one member of the domin to ech member of the rnge. Depending on the function s rule nd its domin, the domin nd rnge might ech consist of finite number of vlues; or either the domin or rnge (or both) might consist of n infinite number of vlues. Emple: Emple: In the function f() = +, if the domin of is the set {,4,6}, then pplying the rule tht f() = + to ll vlues in the domin yields the function s rnge: the set {,5,7}. (All vlues other thn, 4, nd 6 re outside the domin of, while ll vlues other thn, 5, nd 7 re outside the function s rnge.) In the function f() =, if the domin of is the set of ll rel numbers, then pplying the rule tht f() = to ll vlues in the domin yields the function s rnge: the set of ll non-negtive rel numbers. (Any negtive number would be outside the function s rnge.) Eercise 6 Work out ech problem. Circle the letter tht ppers before your nswer.. If f() = +, nd if the domin of is the set {,8,5}, then which of the following sets indictes the rnge of f()? { 4,,,,, 4} {,, 4} {4, 9, 6} {, 8, 5} {ll rel numbers}. If f() = 6 4, nd if the domin of consists of ll rel numbers defined by the inequlity 6 < < 4, then the rnge of f() contins ll of the following members EXCEPT: If the rnge of the function f() = is the set R = {0}, then which of the following sets indictes the lrgest possible domin of? { } {} { } {, } ll rel numbers 4. If f() = 5+ 6, which of the following indictes the set of ll vlues of t which the function is NOT defined? { < } { < < } { < } { < < } { < } 5. If f() =, then the lrgest possible domin of is the set tht includes ll non-zero integers. ll non-negtive rel numbers. ll rel numbers ecept 0. ll positive rel numbers. ll rel numbers.

12 56 Chpter 5 7. LINEAR FUNCTIONS EQUATIONS AND GRAPHS A liner function is function f given by the generl form f() = m + b, in which m nd b re constnts. In lgebric functions, especilly where defining line on the y-plne is involved, the vrible y is often used to represent f(), nd so the generl form becomes y = m + b. In this form, ech -vlue (member of the domin set) cn be pired with its corresponding y-vlue (member of the rnge set) by ppliction of the function. Emple: In the function y = +, if the domin of is the set of ll positive integers less thn 5, then pplying the function over the entire domin of results in the following set of (,y) pirs: S = {(,5), (,8), (,), (4,4)}. In ddition to questions requiring you to solve system of liner equtions (by using either the substitution or ddition-subtrction method), the new SAT includes questions requiring you to recognize ny of the following: A liner function (eqution) tht defines two or more prticulr (,y) pirs (members of the domin set nd corresponding members of the rnge set). These questions sometimes involve rel-life situtions; you my be sked to construct mthemticl model tht defines reltionship between, for emple, the price of product nd the number of units of tht product. The grph of prticulr liner function on the y-plne A liner function tht defines prticulr line on the y-plne. Vritions on the ltter two types of problems my involve determining the slope nd/or y-intercept of line defined by function, or identifying function tht defines given slope nd y-intercept. Emple: In the liner function f, if f( ) = nd if the slope of the grph of f in the y-plne is, wht is the eqution of the grph of f? y = y = + y = 6 y = y = The correct nswer is. In the generl eqution y = m + b, slope (m) is given s. To determine b, substitute for nd for y, then solve for b: = ( ) + b; = b. Only in choice does m = nd b =.

13 Numbers nd Opertions, Algebr, nd Functions 57 Eercise 7 Work out ech problem. Circle the letter tht ppers before your nswer.. XYZ Compny pys its eecutives strting slry of $80,000 per yer. After every two yers of employment, n XYZ eecutive receives slry rise of $,000. Which of the following equtions best defines n XYZ eecutive s slry (S) s function of the number of yers of employment (N) t XYZ? S =, , 000 N S = N + 80,000 S = 80, 000 +, 000 N S =,000N + 80,000 S = 500N + 80,000. In the liner function g, if g(4) = 9 nd g( ) = 6, wht is the y-intercept of the grph of g in the y-plne? If two liner function f nd g hve identicl domins nd rnges, which of the following, ech considered individully, could describe the grphs of f nd g in the y-plne? I. two prllel lines II. two perpendiculr lines III. two verticl lines I only I nd II only II only II nd III only I, II, nd III 4. In the y-plne below, if the scles on both es re the sme, which of the following could be the eqution of function whose grph is l? y = y = + y = + y = y = 5. If h is liner function, nd if h() = nd h(4) =, then h( 0) =

14 58 Chpter 5 8. QUADRATIC FUNCTIONS EQUATIONS AND GRAPHS In Chpter 8, you lerned to solve qudrtic equtions in the generl form + b + c = 0 by fctoring the epression on the left-hnd side of this eqution to find the eqution s two roots the vlues of tht stisfy the eqution. (Remember tht the two roots might be the sme.) The new SAT my lso include questions involving qudrtic functions in the generl form f() = + b + c. (Note tht, b, nd c re constnts nd tht is the only essentil constnt.) In qudrtic functions, especilly where defining grph on the y-plne is involved, the vrible y is often used to represent f(), nd is often used to represent f(y). The grph of qudrtic eqution of the bsic form y = or = y is prbol, which is U-shped curve. The point t which the dependent vrible is t its minimum (or mimum) vlue is the verte. In ech of the following four grphs, the prbol s verte lies t the origin (0,0). Notice tht the grphs re constructed by tbulting nd plotting severl (,y) pirs, nd then connecting the points with smooth curve: The grph of qudrtic eqution of the bsic form = or y = y is hyperbol, which consists of two U- shped curves tht re symmetricl bout prticulr line, clled the is of symmetry. The is of symmetry of the grph of = is the -is, while the is of symmetry in the grph of y = y is the y-is, s the net figure shows. Agin, the grphs re constructed by tbulting nd plotting some (,y) pirs, then connecting the points:

15 Numbers nd Opertions, Algebr, nd Functions 59 The new SAT might include vriety of question types involving qudrtic functions for emple, questions tht sk you to recognize qudrtic eqution tht defines prticulr grph in the y-plne or to identify certin fetures of the grph of qudrtic eqution, or compre two grphs Emple: The grph shown in the y-plne bove could represent which of the following equtions? = y = y y = y = = y The correct nswer is. The eqution y = represents the union of the two equtions y = nd y =. The grph of y = is the prbol etending upwrd from the origin (0,0) in the figure, while the grph of y = is the prbol etending downwrd from the origin.

16 60 Chpter 5 Emple: In the y-plne, the grph of y + = downwrd. upwrd. to the right. to the left. either upwrd or downwrd. shows prbol tht opens The correct nswer is. Plotting three or more points of the grph on the y-plne should show the prbol s orienttion. First, it is helpful to isolte y in the eqution y =. In this eqution, substitute some simple vlues for nd solve for y in ech cse. For emple, substituting 0,, nd for gives us the three (,y) pirs (0, ), (,0), nd (,0). Plotting these three points on the yplne, then connecting them with curved line, suffices to show prbol tht opens upwrd. An SAT question might lso sk you to identify qudrtic eqution tht defines two or more domin members nd the corresponding members of the function s rnge (these questions sometimes involve models of rel-life situtions).

17 Numbers nd Opertions, Algebr, nd Functions 6 Eercise 8 Work out ech problem. Circle the letter tht ppers before your nswer.. Which of the following equtions defines function contining the (,y) pirs (, ), (, 4), (, 9), nd (4, 6)? y = y = y = y = y =. The figure below shows prbol in the yplne. 4. Which of the following is the equtions best defines the grph shown below in the y-plne? y = = y Which of the following equtions does the grph best represent? = (y ) = (y + ) = (y ) y = ( ) + y = ( ). In the y-plne, which of the following is n eqution whose grph is the grph of y = trnslted three units horizontlly nd to the left? y = y = + y = ( ) y = ( + ) y = = y = y y = 5. ABC Compny projects tht it will sell 48,000 units of product X per yer t unit price of $,,000 units per yer t $ per unit, nd,000 units per yer t $4 per unit. Which of the following equtions could define the projected number of units sold per yer (N), s function of price per unit (P)? 48, 000 N = P + 48, 000 N = P 48, 000 N = P , 000 N = P , 000 N = P + 8