9. v > 7.3 mi/h x < 2.5 or x > x between 1350 and 5650 hot dogs

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1 .5 Etra Practice. no solution. (, 0) and ( 9, ). (, ) and (, ). (, 0) and (, 0) 5. no solution. ( , ) and ( 5 5 5, ) 7. (0, ) and (, 0). (, ) and (, 0) 9. (, 0) 0. no solution. (, 5). a. Sample answer: elimination; Both equations are quadratic. b. Sample answer: es; Substitution would be used because one equation is linear.. and ; Let be the irst number and be the second number. The two equations + = 5 and + = 7 can be solved usin substitution to ind both numbers.. Eplorations. The -values or which the raph touches or is below the -ais are solutions;. a. A; < or > b. C; c. E; < < d. B; or e. F; < <. D; < or >. Rewrite the inequalit so that one side is 0. Graph the related quadratic unction to see where the raph is above or below the -ais.. Use the raph to determine or what -values the unction is positive and neative. a. < or > b. < < c. or. Etra Practice. A; The -intercepts are = and =. The test point (, 5) does not satis the inequalit.. C; The -intercepts are = and =. The test point (, 5) satisies the inequalit.. D; The raph has no -intercepts. The test point (, 5) satisies the inequalit.. B; The raph has no -intercepts. The test point (, 5) does not satis the inequalit v > 7. mi/h <.5 or > between 50 and 550 hot dos 5 9 Chapter Maintainin Mathematical Proicienc. 7. r h π t 7. t. 0 m mm 0. π t.57 t. units. Eplorations a. A; The unction has -intercepts o, 0, and. Substitutin a lare positive value into the unction results in a lare positive value, and substitutin a lare neative value into the unction results in a lare neative value. The raph displas this because it rises to the riht and alls to the let. b. D; The unction has -intercepts o, 0, and. It rises to the let and alls to the riht. c. B; The unction has a -intercept o, and -intercepts o and. It alls both to the riht and let. d. F; The raph has an - and -intercept o 0. It rises both to the let and riht. e. E; The raph has an - and -intercept o 0. It alls to the let and rises to the riht.. C; The raph has -intercepts o, 0, and. It rises both to the let and riht. A Alebra Copriht Bi Ideas Learnin, LLC

2 . a. The -intercepts are, 0, and ; Sample answer: The can be veriied b substitutin the values into the unction to ind that ( ) = 0, (0) = 0, and () = 0. b. The -intercepts are, 0, and ; Sample answer: The can be veriied b substitutin the values into the unction to ind that ( ) = 0, (0) = 0, and () = 0. c. The -intercepts are and ; Sample answer: The can be veriied b substitutin the values into the unction to ind that ( ) = 0 and () = 0. d. The -intercept is 0; Sample answer: It can be veriied b substitutin the value into the unction to ind that (0) = 0. e. The -intercept is 0; Sample answer: It can be veriied b substitutin the value into the unction to ind that (0) = 0.. The -intercepts are, 0, and ; Sample answer: The can be veriied b substitutin the values into the unction to ind that ( ) = 0, (0) = 0, and () = 0.. The parent unction alwas oes throuh the oriin. The cubic unction alls on one side and rises on the other. The quartic unction either rises on both sides or alls on both sides.. a. true; This means that a n < 0, so the unction increases as approaches neative ininit and decreases as approaches positive ininit. b. alse; A quartic unction alls to the let and alls to the riht when a n < 0, and rises to the let and rises to the riht when a n > 0.. Etra Practice. polnomial unction; () = ; deree: (quartic), leadin coeicient:. not a polnomial unction. polnomial unction; () = ; deree: (linear), leadin coeicient: 5. polnomial unction; p() = + ; deree: (quadratic), leadin coeicient: () as and () as. () as and () as 9. The deree o unction is odd and the leadin coeicient is neative The deree is even and the leadin coeicient is neative.. a From 00 until mid-ear 0, the number o students raduatin in our ears increased. Around mid-ear 0, the number o students raduatin in our ears started decreasin. b. The averae rate o chane is about 50 less students who are raduatin in our ears ever ear. c. Because the raph declines so sharpl in the ears ater 0, it is most likel not accurate. The model ma be valid or a ew ears beore 0, but in the lon run, decline ma not be reasonable. Ater 0, the number o students raduatin collee becomes neative. Because neative values do not make sense iven the contet, the model cannot be used or ears ater 0.. Eplorations. a. ( + + ); ; b. (a + ab + b ); a + a b + ab + a b + ab + b ; a + a b + ab + b c. ( + ); + + ; + d. (a ab + b ); a a b + ab a b + ab b ; a a b + ab b. a. The absolute values o the coeicients are equal to the rd row o Pascal s Trianle (when countin the top row as the 0th row.) b. For the irst term o the binomial, the eponents bein at and decrease b one with each term. For the second term o the binomial, the eponents bein at 0 and increase b one with each term. c. Because the eponent is a, the irst term o the product will be the irst term o the binomial cubed, times the second term o the binomial raised to the 0th power, times the coeicient. The second term will be the irst term o the binomial squared, times the second term o the binomial raised to the st power, times the coeicient, and so on; For the binomial ( + ), the pattern is For the binomial ( ), the pattern is a b. + c d e Etra Practice v v + v Copriht Bi Ideas Learnin, LLC Alebra A9

3 0. 5t 5t + t t Eplorations. a. F; The raph has -intercepts o, 0, and. Solvin or () results in a unction with the same -intercepts. b. C; The raph has -intercepts o and. Solvin or () results in a unction with the same -intercepts. c. A; The raph has -intercepts o,, and. Solvin or () results in a unction with the same -intercepts. d. D; The raph has -intercepts o,, and. Solvin or () results in a unction with the same -intercepts. e. E; The raph has -intercepts o and. Solvin or () results in a unction with the same -intercepts.. B; The raph has -intercepts o,, and. Solvin or () results in a unction with the same -intercepts.. a. + b. + c. + d. + e I ou divide the polnomial b one o its actors, then the result is the remainin actors.. Etra Practice A; () () = so the remainder must be. 0. D; () () + = 0 so the remainder must be 0.. B; () + () = so the remainder must be.. C; () + () + = so the remainder must be.. () =. ( ) = 5. ( 5) = 50. () = 7 7. k =. Eplorations. a. C; ( + )( + ) = 0 b. F; ( + )( )( ) = 0 c. E; ( + )( ) = 0 d. A; ( + )( ) = 0 e. D; ( + )( + )( )( ) = 0. B; ( + )( )( ) = 0 You can write each actor as a, where a is an -intercept, because substitutin a or makes the related polnomial unction 0.. a. () = ( )( + ); The -intercepts o the raph are and. b. () = ( )( + ); The -intercepts o the raph are 0,, and. c. () = ( )( + ); The -intercepts o the raph are 0,, and. d. () = ( + )( )( ); The -intercepts o the raph are,, and. e. () = ( + )( + )( ); The -intercepts o the raph are 0,,, and.. () = ( + )( + )( )( ); The -intercepts o the raph are,,, and.. B lookin at the raph o a polnomial, the -intercepts can be ound and then used to write the equation in actored orm.. -intercepts. Etra Practice. 0( )( 5). m(m + )(m ) ( m + 9 ). (a + b) ( 9a ab + b ). t (5t + )(t + )(t ) 5. ( + )( ) ( + ). 5( p ) ( p + ) 7. 0(k + )(k )(9k + ). (a + )(a + a + )(a ) 9. ( 7)( + ) 0. (5z )(z + ). ( 5)( + ). m ( m)( + m) ( + m ). ( ). 5m (m 7) () = ( + )( )( + ) () = ( )( + )( + ) () = ( 5)( )( + ).5 Eplorations. a. D; =, =, and = ; The raph appears lat as it crosses the -ais at the repeated solution o =. b. C; =, =, and = ; The raph appears lat as it crosses the -ais at the repeated solution o =. c. A; =, =, and = ; At the repeated solution o =, the raph touches the -ais, but does not cross it. d. E; =, = 0, and = e. B; =, =, and = ; At the repeated solution o =, the raph touches the -ais, but does not cross it.. F; = 0, =, and =. a. The equation has a repeated solution; At =, the raph touches the -ais, but does not cross it; = 0, =, =, and = b. The equation does not have a repeated solution; It crosses the -ais at each -intercept; =, = 0, =, and = A0 Alebra Copriht Bi Ideas Learnin, LLC

4 c. The equation has two repeated solutions; At = 0 and at =, the raph touches the -ais, but does not cross it; = 0, = 0, =, and = d. The equation has a repeated solution; At = 0, the raph appears lat as it crosses the -ais; = 0, = 0, = 0, and =. A polnomial unction has repeated solutions i its raph touches the -ais at a point without crossin it, or the raph appears lat at an -intercept.. Sample answer: + = ( )( + )( + ) = 0.5 Etra Practice. r = 0, r =, and r =. = 0, =, and =. m = 0, m = 5, and m = 5. =, =, =, and = 5. =, =, and =. c = 5 and c = 5 7. = 0, =, and =. = 0 and = (, 0) (0, 0) (, 0) =, =, and = 0 (.5, 0) (, 0) 0 0. = and = (, 0) (, 0) (.5, 0). D.,.,,.. () = a. = 0 and = b. () = ( + ). Eplorations. a. D; = and = ± i b. E; =, =, and = c. B; =, =, and = (0, 0) (, 0 ) d. F; = and = ± i e. A; = and = ±i. C; = 0, =, and = A cubic unction alwas has solutions. I the raph crosses the -ais three times, then all three solutions are real. I the raph crosses the -ais onl once, then one solution is real and the other two are imainar or there is a repeated real solution.. a. no imainar solutions; The raph and table o values show that it has -intercepts; =, = 0, =, and = b. two imainar solutions; The raph and table o values show that it has -intercepts; =, =, and = ±i c. two imainar solutions; The raph and table o values show that it has -intercepts; =, =, and = ± i d. no imainar solutions; The raph shows that it has -intercepts, includin one repeated solution; =, =, =, and =. The deree o the unction ives the number o solutions. I the raph or table o values show a number o real solutions that is less than the deree b an even number, takin into account repeated solutions, then the remainin solutions are imainar.. no; Because the end behaviors o a cubic unction are alwas opposite, the unction must cross the -ais in at least one place.. Etra Practice.,, + i, and i. 5, + i, and i. 7, 7, i, and i. 5, + i, and i 5. () = 5 +. () = () = () = Eplorations. a. () = ( ) b. () = c. () = + d. () = ( + ). a. () = ( ) b. () = ( ) +. horizontall, verticall, in the -ais, in the -ais. The raph o is a translation one unit let and three units up o the raph o. Copriht Bi Ideas Learnin, LLC Alebra A

5 .7 Etra Practice. The raph o is a translation 9 units down o the raph o. 5. The raph o is a vertical shrink b a actor o, ollowed b a relection in the -ais o the raph o.. The raph o is a translation unit let and units up o the raph o.. The raph o is a translation 0 units riht, ollowed b a translation unit up o the raph o.. The raph o is a vertical stretch b a actor o 5 and a translation units riht, ollowed b a relection in the -ais o the raph o. 7. () (). () = (). The raph o is a horizontal stretch b a actor o, ollowed b a translation units down o the raph o. () The raph o is a relection in the -ais, ollowed b a translation 9 units down o the raph o. 9. () = () () The raph o is a vertical stretch b a actor o 5 o the raph o. 0. () = ( + ) + ( + ) 0. () = A Alebra Copriht Bi Ideas Learnin, LLC

6 . Eplorations. a. C; is a quadratic unction with a -intercept o ; (0.75, 5.) b. F; is a quadratic unction with a -intercept o ; (.50, 0.5) c. A; is a cubic unction with a -intercept o ; (.5,.), ( 0., 0.5) d. D; is a cubic unction with a -intercept o ; (.9,.0), (.9,.0) e. B; is a quartic unction; (, ), (0.7, 0.5), (., 5.). E; is a ith deree unction; ( 0.9,.), (0.77, 5.7). The raph o ever polnomial unction o deree n has at most (n ) turnin points.. es; () = has no turnin points because its raph is alwas increasin.. Etra Practice. ( 0.0, 5.0).. (.77,.) (, 0) ( 0., 0) (., 0) (.,.5) The -intercepts o the raph are =, 0., and.. The unction has a local maimum at (.77,.) and a local minimum at (.,.5); The unction is increasin when <.77 and when >. and decreasin when.77 < <.. The unction is neither even nor odd. ( 0.,.09) (., 0) (., 0) (0.,.09) (0, 0). (, 0) (.5, 0) (.0,.0) (.5, 0) The -intercepts o the raph are =, =.5, and =.5. The unction has a local maimum at ( 0.0, 5.0) and a local minimum at (.0,.0); The unction is increasin when < 0.0 and when >.0 and decreasin when 0.0 < <.0; The unction is neither even nor odd. ( 0., 0) (.5, 0) (.5, 0) 0 (.,.) ( 0.,.) 0 0 (.,.7) (0, 0) The -intercepts o the raph are = 0,.5, 0. and =.5. The unction has a local maimum at ( 0.,.) and local minimums at (.,.7) and (.,.); The unction is increasin when >. and when. < < 0., and decreasin when <. and 0. < <.; The unction is neither even nor odd. 5.. The -intercepts o the raph are., = 0, and.. The unction has a local maimum at ( 0.,.09) and a local minimum at (0.,.09); The unction is increasin when < 0. and when > 0. and decreasin when 0. < < 0.; The unction is odd. (, 0) ( 5, 0) 00 (., 0.5) (, 0) (0, 00) (5, 0) (., 0.5) The -intercepts o the raph are = 5, =, =, and = 5. The unction has a local maimum at (0, 00) and local minimums at (., 0.5) and (., 0.5); The unction is increasin when. < < 0 and when >. and decreasin when <. and 0 < <.. The unction is even. (, 0) (, 0) ( 0, ) The -intercepts o the raph are = and =. The unction has a local maimum at (, 0) and a local minimum at ( 0, ); The unction is increasin when 0 < < and decreasin when < 0 and when > ; The unction is neither even nor odd. Copriht Bi Ideas Learnin, LLC Alebra A

7 .9 Eplorations. a., 7,, 9, 0,, ;,,,,, ; quadratic; The second dierences are constant. b. quadratic; = c. 0 t d. about 0 mph. Determine the deree o the unction b subtractin until constant dierences are ound. Then use technolo to ind the model that best its the data.. The model its the data perectl; no; For speeds less than miles per hour, the distance is neative, and speeds beond 5 miles per hour are not realistic..9 Etra Practice. () = ( ) ( + ). () = ( + ). () = ( + ) ( + ). () = ( + )( )( ) 5. ; () = +. ; () = ; () =. ; () = = + 5;,9,000 bacteria 0. = + ; 007 Chapter 5 Maintainin Mathematical Proicienc. c 0. q. 5.. d m0. n 7. = +. = = 0. = =. = +. ( + + ) = ( ) = ( ) ; Appl the Order o Operations and combine the like terms in parentheses irst. Then square the individual actors in the numerator and the denominator. 5. Eplorations. a. 9 = = 9 / b.. / c. = = / d.. / e. = = /..9 /. a. 5 /. b. / =.00 c. 9 /. d. 0 /5. e. 5 / / =.00. a. ( ) =.00 b. ( ) 5. c. ( ).5 d. ( 0 ). e. ( ) =.00. ( 5 0 ).. Use the inde o the radical as the denominator and the eponent placed on the radical as the numerator. 5. a. ; The square root o is and is. b. ; The ith root o is and is. c. 5; The ourth root o 5 is 5 and 5 is 5. d. ; The square root o 9 is 7 and 7 is. e. 5; The cube root o 5 is 5 and 5 is 5.. 0,000; The eponent reduces to and 00 is 0, Etra Practice. 5. no real square roots. and C; The denominator o the eponent is and the numerator is.. D; The denominator o the eponent is and the numerator is.. B; The denominator o the eponent is and the numerator is. 5. A; The denominator o the eponent is and the eponent is neative.. = 7.. and 7.. = 9. = 0 0. in. in. in. 5. Eplorations. a. a ; D b. a b ; C c. a ; B d. a 7 ; A a e. b ; G. a ; F. ; E. a. 5 b. c. d. 00 e. 7.. a. ; b. 5 5 ; 5 c. 7 ; d. 9 ; 7 A Alebra Copriht Bi Ideas Learnin, LLC