. + 7i. 0 i 7. 9 i. 79 + i 9. 7 i 0. 7 i. + i. 0 + i. a. lb b. $0. c. about $0.9 Chapter. Start Thinkin () = () = The raph o = is a curv line that is movin upward rom let to riht as increases. The raph o = is similar to a parabola that opens up with its verte at the oriin. Both raphs have positive -values when is positive. When is neative, the are neative, and the -values o -values o ( ) are positive; The eponent is even; es; ( 0, 0 ) and (, ). Warm Up. 0. 9. 0. 0....9. Cumulative Review Warm Up. down. up. down. up. Practice A. polnomial unction; = + 7, deree is, cubic, leadin coeicient is. not a polnomial unction. polnomial unction; = + + 0, deree is, quartic, leadin coeicient is = +, deree is, quadratic, leadin coeicient is. polnomial unction;.. 7.,. + as + and. 9. h as + and h. 0.... + as + as. is increasin when >. is decreasin when <. is positive when < and >. is neative when < <.. is increasin when <. and >.. is decreasin when. < <.. is positive when < < 0 and >. is neative when < and 0 < <.. The deree is even and the leadin coeicient is neative.. Practice B. not a polnomial unction = + 7, deree is, quadratic, leadin coeicient is. polnomial unction;. polnomial unction; = +, deree is, quartic, leadin coeicient is. not a polnomial unction q() = k() = +.. 7 7. 9 h() = + () = + Copriht Bi Ideas Learnin, LLC Alebra A
. as + and. as. 9. h + as + and h. as 0. q() = The deree is odd and the leadin coeicient is positive.. Sample answer: ; 0 0 h. h() = 0 0 0. Enrichment and Etension. + +.7. 0. 0.7.. k() = + () = + + +... + + 0..9. 7.9 + + 0.9.07.9.9 0.0.9 0 0.0 +.7.7 + 99.7. Puzzle Time COBWEBS.. Start Thinkin Sample answer: The deree is even and the leadin coeicient is neative. ( )( ) + = + = ; ( )( ) + = + 9 = 9; es; In each eample, the middle terms cancel out, leavin onl two terms. The irst term is the square o the irst term in each binomial. The second term is the square o the second term in each binomial; + + = + + + = + + ; = + 9 = + 9; no; The sins are the same inside the binomials, so the middle terms no loner cancel. A Alebra Copriht Bi Ideas Learnin, LLC
. Warm Up. 7. + 7. m +. 0r. z z. m. Cumulative Review Warm Up. ; +. ; ( w ).. ; ( ).. Practice A...... 7. + 7 9 7 + + + 9 + + + +. 9; ( z + ) ; 79 7 ; s + 7 + + + 7 + 9. + + 0 9. 0.. 0 + + 7 + + 9 0. The neative was distributed incorrectl; + 7 = + p. Practice B. 7 + + +. 7 +.. + + + 0 9 9 + 7 + 0 + + +.. 7.. + + 0 9 7 + 0 + + + 0 9. The eponents were multiplied instead o added; + 7 = + 0....... + 7 + + 0 + + 0 9 t + t + 9 pq + pq+ + 7. a. Sample answer: + + 9 b.. Enrichment and Etension. a =, b =, c =. a =, b =, c =, d =... + + + 0 + 9. a = 0, b =, c =, d = 0. a =, b =, c =, d =. a = 7, b = 0, c =. 7 0 7.. + +. a = 9, b = 0, c = or a = 9, b = 0, c = 9. p p + 9 Copriht Bi Ideas Learnin, LLC Alebra A
7. 0 0 0 + = + + + 0 +. + + = + 90 9. 0 + 90 + = + 0 + 0.. +. 0 = + 0 0 + 0 9 + bc + de = b c + b c de + b c d e +. 0bcde + bcd e + bcde + de. Puzzle Time A WALKIE TALKIE. Start Thinkin ( )( ) + ; Inverse operations undo one another, so i two binomials are multiplied to make a product, ou can divide the product b one binomial to obtain the other binomial; no; Factorin will onl work as division i there is no remainder. It is possible to divide polnomials that are not actorable.. Practice A. + +. + +. 7. 9. + 9 + + + +. k = ; 9 + 0 0 7 9. + +. +.. 0. + + 7 + + + + + 7 = + + +..... + + ; Multipl the result b +.. Practice B. + +.. + 9 + + + 0 7 + 90. Warm Up. ( t + ). k ( k ). 7 7. ab ( ab a + b ). ( + )( ). ( n )( n ). ( + 7)( + ). Cumulative Review Warm Up. =. = +. = +. + +. + + 7. 7 + 7 +. 0 + 0 + 9 + A Alebra Copriht Bi Ideas Learnin, LLC
9. 0 + 0 + + 0. The powers in the quotient are too lare b. The + ; remainder ( ) was not divided b 0 7 9 + + 7 = + + +.. = + = 0. (0.,.) =. = +. 7. 0. 0. 9. k =. Enrichment and Etension. =. = +. = +. = +. (.,.). =. = a + b ad. Puzzle Time HE WAS ALWAYS WILLING TO LEND AN EAR (, ) = = +. Start Thinkin es; You can roup the terms with coeicients o and toether, or ou can roup the terms with coeicients o and 0 toether; es; You can roup and toether, and ou can roup and 0 toether.. Warm Up.. rs.. z. ab.. Cumulative Review Warm Up. = 0... (0.,.7) = + = 0. = + + ( 0.7, 0.) = 0.7 ( 0., ) = +. Practice A. ( )( + ). 9p ( p + )( p ). n ( n )( n ). k ( k + )( k ). w ( w + )( w ). q ( q + )( q 7 ) Copriht Bi Ideas Learnin, LLC Alebra A7
7. ( + )( + 9). ( + 0)( 0 + 00) 9. ( w )( w + w + ) 0. ( )( + ). ( q )( q + 9) + +. ( )( )( + ). ( d )( d ). ( p )( p + ). ( n + )( n + 7). ( + )( + )( ) 7. no. es 9. es 0. no. Sample answer: = + ; ( ) ( )( + ) = + k =. 9; = 7 9 + ; ; 7 9 + = +. ( 9 + )( + )( ). t ( t + )( t ). v ( v )( v ) 7. es. no 9. no 0. no = + 7 ;. Sample answer: ( + ; ) + = + 7. a. ( ) + = ; ( hk ) = ( ) r =,,0, hk, =,0, r = b. + = ;. a. ( a + b )( c d) + n b.. Practice B. t ( t + )( t ). p ( p 7)( p ). ( + )( ). a ( a + 9 )( a ). j 7 ( j )( j ). q ( q + )( q + ) 7. p ( p )( p + p + ). k ( k + )( k k + ) 9. w ( w )( 9w + w + ) 0. ( 7)( + ). ( m )( m + )( m ). ( w )( w + )( w ). ( s + )( s + )( s ) c. ( ) ( ) ( hk, ) = (, ), r = + = ;. Enrichment and Etension. 7..... = + + + 7. ( + ) A Alebra Copriht Bi Ideas Learnin, LLC
7 7 a + b = a + b a a b + a b a b + ab ab + b ). ( = + = 7 7 7 7 9. ( a b ) ( a b )( a b ) ( a + b)( a b)( a a b + a b a b + )( ab ab + b a + ab+ ab + ab + ab + ab + b ) 0 0 0. ( ) = ( + )( ) = ( + )( )( + + ) ( + + + + ). Puzzle Time QUARTERBACK. Start Thinkin. k =, k = 0. = 0, =. n =, n =, n =. p = 0, p = ± 7. u = 0, u = ± 7. =, = 0, = 0 0. =, = () = + () = + 0 () = 0 ( 0, 0) and, 0 ; The unction simpliies to 0 = 0; These points have -values which ield a -value o zero, meanin the raph crosses the -ais. These are the onl points that can be inserted into the unction = to et this result.. Warm Up. 0 t =. =. r = 0. z = 0. m =. b = 0. Cumulative Review Warm Up. < or >. 9. < or > 9. =, = 0, = h() = 00 00 00 00 0. =, =, = 0 () = 0 0. < <. < or >. < < 0 0. Practice A. q =, q = 0, q =. C Copriht Bi Ideas Learnin, LLC Alebra A9
. The actors o include ± and ± ; = + ; Possible zeros: ±, ±, ±, ±, ± 9, ±. =, = 0. =, =, = () = 9 + 0. =, =, = = + 9 9;. Sample answer: 0; = = 0. a. k = b. k =. Practice B.. =, = 0. h = 0, h = ± q = ±. w = ±. p =, p =±. = ±, = 7. =, = 0, = () = + 0 + 0 00 00. =, = 0, = () = + 0 00. B. The actors o include ± ; = + ; Possible zeros: ±, ±, ±, ±, ±, ±. = 0. =, =, =. Sample answer: = 0 9 + 9; 7 0; = 7 = 0. heiht is cm, side lenth is 9 cm. Enrichment and Etension P = 7 7+.. P = + =, a can be an real number. P a ( ) 00 P = + + 0. 9. =, =, = h() = + 0 0 P = + +.. Puzzle Time IT WAS A BREEZE WITH ONLY A FEW FOGGY PATCHES 0 0 A0 Alebra Copriht Bi Ideas Learnin, LLC
. Start Thinkin Sample answer: Function Number o -intercepts = +. () = () = + = The deree o the unction and the number o - intercepts are the same; no; Sometimes, there are solutions to polnomial unctions that are imainar numbers, which are not shown on the raph o the unction.. Warm Up....... Cumulative Review Warm Up. h = + j = + () = () = The raph o = is a vertical translation units down o the parent linear unction.. The raph o = + is a horizontal translation units let o the parent absolute value unction. h() = () = Sample answer: The raph o h = is a relection in the -ais o the parent linear unction.. Practice A.....,,, i i.,,, 7. 0,, i, i.,,, 9. ; The raph shows real zero, so the remainin zeros must be imainar. 0. 0; There are zeros or this unction. The raph crosses the -ais twice and touches the -ais once at the repeated zero, leavin 0 imainar zeros... = + 0 + () = = + +. = +. The raph o = is the parent quadratic unction, so there was no transormation.. Sample answer: = ; Because i is a zero, i is also a zero. The raph touches the -ais at (has a multiplicit o ) and the raph crosses the -ais at. Copriht Bi Ideas Learnin, LLC Alebra A
.. 7.. Practice B.....,,,.,,, 7.,,,,.,,,, 9. ; Sample answer: There are our zeros or this unction. The raph crosses the -ais twice, leavin two imainar zeros. 0. ; Sample answer: There are three zeros or this unction. The raph crosses the -ais onl once, leavin two imainar zeros. = + 0. = + +. = 7 +.. Comple zeros come in pairs, so the remainin zero cannot be comple.. C Positive real zeros Neative real zeros Imainar zeros Total zeros 0 0 Positive real zeros Positive real zeros Neative real zeros Neative real zeros Imainar zeros Imainar zeros Total zeros 0 0 0 Total zeros. Enrichment and Etension. a and b. = ( + )( + )( + )( ) c.,,,. a. = ( + )( )( + ) b. = ( + )( ) ( ) ( + ) c.,,, +. a. = ( )( )( + ) b. = ( )( ) ( + i) ( i) c.,,, i + i. a. = ( + )( + )( + ) b. = ( + )( + ) ( + ) ( ) c.,,, +. a. = ( + )( )( ) b. = ( + )( )( + )( ) c.,,,. a. = ( )( + )( )( + 9) b. = ( )( + )( )( + i) ( i) c., i,,. Puzzle Time A PUP TENT A Alebra Copriht Bi Ideas Learnin, LLC
.7 Start Thinkin Function Transormation Function = Translation = units riht = ( + ) = = + Translation units let Translation units down Translation units up Transormation Relection in the -ais = Horizontal shrink b a actor o The transormations o = behave in the same manner as other parent unction transormations. Numbers added or subtracted inside parentheses translate the raph let or riht, and numbers added or subtracted outside the parentheses translate the raph up or down. Numbers multiplied inside the parentheses horizontall stretch or shrink the raph, and numbers multiplied outside the parentheses verticall stretch or shrink the raph..7 Warm Up. Sample answer: The raph o is a vertical stretch b a actor o o the raph o the parent unction.. The raph o h is a relection in the -ais, ollowed b a vertical shrink b a actor o o the raph o the parent unction.. Sample answer: The raph o is a vertical stretch b a actor o o the raph o the parent unction.. The raph o is a relection in the -ais, ollowed b a vertical shrink b a actor o o the raph o the parent unction. = Horizontal stretch b a actor o = Vertical stretch b a actor o.7 Cumulative Review Warm Up. The raph o = + is a vertical translation units up o the raph o the parent quadratic unction. () = + = is a horizontal translation unit riht o the raph o the parent quadratic unction.. The raph o () = () = = + is a horizontal translation units let o the raph o the parent quadratic unction.. The raph o () = ( + ) = + is a horizontal translation units riht, ollowed b a vertical translation units up o the raph o the parent quadratic unction.. The raph o () = ( ) () = () = ( ) + () = Copriht Bi Ideas Learnin, LLC Alebra A
.7 Practice A. The raph o is a vertical translation units down o the raph o the parent unction.. The raph o is a vertical shrink b a actor o, ollowed b a vertical translation units down o the raph o the parent unction.. The raph o is a horizontal translation units let o the raph o the parent unction.. The raph o is a vertical shrink b a actor o, ollowed b a horizontal translation units riht o the raph o the parent unction.. The raph o is a relection in -ais, ollowed b a vertical stretch b a actor o o the raph o the parent unction. = + 7.. The raph o is a vertical stretch b a actor o, ollowed b a vertical translation units down o the raph o the parent unction. The raph o is a horizontal translation unit riht o the raph o the parent unction. = +. The raph o is a vertical stretch b a actor o o the raph o the parent unction. A Alebra Copriht Bi Ideas Learnin, LLC
9. The parent unction was translated units down instead o units up.. The raph o is a vertical stretch b a actor o, ollowed b a vertical translation units up o the raph o the parent unction. = + 0. =. W = 7 + +. a. Z =, + + b..7 Practice B. The raph o is a horizontal translation units riht, ollowed b a vertical translation units down o the raph o the parent unction.. The raph o is a horizontal translation unit riht, ollowed b a vertical translation units up o the raph o the parent unction.. The raph o is a vertical shrink b a actor o, ollowed b a vertical translation units down o the raph o the parent unction.. The raph o is a vertical shrink b a actor o, ollowed b a horizontal translation units let o the raph o the parent unction. = + 7.. The raph o is a relection in the -ais, ollowed b a vertical stretch b a actor o o the raph o the parent unction. The raph o is a relection in the -ais, ollowed b a vertical shrink b a actor o o the raph o the parent unction. Copriht Bi Ideas Learnin, LLC Alebra A
. = +. Start Thinkin 0 0 The raph o is a relection in the -ais, ollowed b a vertical translation units up o the raph o the parent unction. 9. The raph o is a vertical stretch b a actor o, not a vertical shrink b a actor o, o the raph o the parent unction = ; The raph o is a vertical stretch b a actor o, ollowed b a translation units up o the raph o. = + 7 0. = + + +. = + 9; = 9.; When = inches, the volume o the bo is 9. cubic eet. Z = + 9,. a. W W b..7 Enrichment and Etension = ( + ) ; ( ) ( ) = + ; = + ; ; = ; = = ( ) + ; = ( ) ; = ( ) = ( ) ; ( ).7 Puzzle Time FRYDAY = + ; The shape o the raph o the unction is a rounded N ; There is one zero where < <, another zero where < < 0, and a third zero where < < ; no; The -values are chanin sins, but the table does not show an -value when = 0.. Warm Up. (, ). (, ). (, 9).,. (, ). (, 0 ). Cumulative Review Warm Up. = ( + ) 9; (, 9). = ( + ) ; (, ). = ( ) ; (, ). h = ( 0) 9; ( 0, 9). h = ( ) + ; (,). = ( ) ; (, ). Practice A... () = ( + ) ( ) 0 h() = ( )( )( + ) () = ( ) ( + )( + ) A Alebra Copriht Bi Ideas Learnin, LLC
. () = ( ) ( + ). h() = +. The unction was raphed as i the zero = had a multiplicit o instead o the zero =. () = ( ) ( + ). -intercepts:.9, 0; local maimum: none; local minimum: (.,. ); increasin: >.; decreasin: <. () = +.,, 7. 7,,. 9. 0.. () = +,, 0, ; -intercepts: 0.9,,.9; local maimum: local minimum: (.7,. ); increasin: < 0, >.7; decreasin: 0 < <.7 () = + -intercepts:., 0.,,.7; local maimum: ( 0.7, 0. ); local minimum: (.,.7 ); < < 0.9,.0, increasin: 0.9 0.7, >.; decreasin: < 0.9, 0.7 < <.. ( 0.,.09 ), ( 0.,.09 ); The point ( 0.,.09) ( 0.,.09) is a local maimum. The point is a local minimum; The real zeros are., 0, and.; The minimum deree is.. (.0,.9 ); The point.0,.9 is a local maimum; The real zeros are 0.9 and.9; The minimum deree is.. Practice B. () = ( + ) ( ). () = ( )( + )( ) -intercepts: 0,.; local maimum: ( 0.790,.9 ); local minimum: none; increasin: < 0.79; decreasin: > 0.79 0 0 0 0 Copriht Bi Ideas Learnin, LLC Alebra A7
.. h() = ( )( )( + ) () = ( )( + + ). h() = 9 0 0. The unction was raphed as i the zero = 0 had a multiplicit o instead o a multiplicit o... 0. 0 7.,, 9. () = ( + ) () = 0. +.,,. -intercepts:., 0., ; local maimum: ( 0.9,.0 ); local minimum: ( ).,. ; increasin: < 0.9,. < ; decreasin: 0.9 < <. () = + + 0 0 0 -intercepts: 0.7,; local maimum: none; local minimum: ( 0.,.07 ); decreasin: < 0.. a. about. in. b. about 7. in. increasin: 0. < ; c. lenth. in., width. in., heiht. in.. Enrichment and Etension. =.. =.. =.. =.99. =.9. -intercepts: 0.7, 0.7,.9; local maimum: ( 0,. ); local minimum:,. ; increasin: < 0, > ; decreasin: 0 < < () = 0.. Puzzle Time SHIFTY.9 Start Thinkin Sample answer: -intercepts:.7, 0,.7; local maimum: (.,. ); local minimum: ( ).,. ; increasin: <., >.; decreasin:. < <. quadratic; parabola; cubic A Alebra Copriht Bi Ideas Learnin, LLC
.9 Warm Up.....9 Cumulative Review Warm Up. (, ). (, ).9 Practice A = +. = + +. ; = +. ; = + +. ; = + +. 7 = +. a. b. es; The cumulative number o customers will continue to increase..9 Practice B. = + + =. ; =. ; = + 9. 7 9 9 ; = + +. 7 = +. a. b. no; The wave heiht will decrease eventuall. Ater seconds, the heiht o the wave will not be 97 inches..9 Enrichment and Etension. arithmetic; d =. not arithmetic. arithmetic; d = 9. not arithmetic. arithmetic; d =. not arithmetic 7. t = n. t = n n 9. t = n + 9 0. t = n +. n tn = n +. tn =.n + 9.9.9 Puzzle Time ANCHOR Cumulative Review. 7j. p. q +. m +. b + 9. + 7. c +. r + 9. z 0. 9a. 9. 9d 0. =. =. b =. m = 7. a =. p = 9. w = 0. =. =. s =. =. =. =. n n = 9 7. = 9. = 9. a. = +.0 b. $ c. $.0 d. when ou are downloadin sons or ewer or when ou are downloadin 7 sons or more 0. + 7. n. m 9. c + Copriht Bi Ideas Learnin, LLC Alebra A9
. 7. + p. j 7. t m. ( + )( + ) 9. ( )( + ). = 7 ± 7.. ± = 7. = = 9 ± 97 9 ± 0 0. ( + )( 7). ( + )( + ). ( 7)( ). ( 0)( ). ( + 7)( + ). ( )( 9 + ). ( )( + ) 7. ( + )( + ). ( )( + ) 9. ( + 7)( ). 9 ± = 9. 70. 9 + i and 9 i 7. and 7. + and = ± 0. a. t b. $9.0 7. + 0 and 0. a. p ( ) ( ) = + + or p = b. a ( )( ) = + or a = 9 7. + i and i c. lenth = t, width = t d. t. = and =. = and = 9 7. 7 + 0 7. and 7 0 + i 9 i 9 and 0 0. = 7 and =. =. = and = 77. 9 + 7 7. a. student b. students and 9 7 7. = and = 7. 9. 0.. 9 = and = 7 = and = = and = = and = 9 79. a. students b. students 0. The raph o is a translation units riht o the raph o.. The raph o is a translation units up o the raph o.. The raph o is a translation units riht and units up o the raph o.. The raph o is a translation unit let and units down o the raph o.. 7 ± 7 =. = ± 7. The raph o is a relection in the -ais, ollowed b a translation units down o the raph o. A0 Alebra Copriht Bi Ideas Learnin, LLC
. The raph o is a relection in the -ais, ollowed b a translation unit riht and units up o the raph o.. The raph o is a relection in the -ais, ollowed b a vertical stretch b a actor o and a translation units down o the raph o. 7. The raph o is a vertical shrink b a actor o, ollowed b a translation units let and units down o the raph o.. = + 9. = + + 90. = + 9. = + + 9. = + = + + 9. 9. a. t b. t c. t; The ball was initiall hit rom eet above the round. d. maimum e. (.7, 7. );.7 sec Chapter. Start Thinkin Eample = = the are reciprocals. ; Because, Epanded Form Simplest Form + + ( ) ( ). Warm Up. k.. 0 79 z. 0 u v. 9 h j.. Cumulative Review Warm Up = 0.0 +. = + +.. Practice A 0 a b c 9.. ± 7. ±... 7.. 00 9. 0. 7.. 0...7. 9,09. 0.0..0 in. 7.. m. = ± 9. = 0. =., 0.. =. =.7. = ±.7..7%. Practice B. 7. no real roots.... 7.. 0 9. 0. 0.. 7..,9.0. 0.07..... 7. in. 7. 7 m. = ±.7 9. =.97 0. = ±. =.. = ±. = 7. a. about 0.7 au b. ears. Enrichment and Etension. n =. n =. n =. n =. n = or n =. n = Copriht Bi Ideas Learnin, LLC Alebra A