. Midterm Review Integrated Math 3 1. Which of the following ordered pairs is in the solution set of the system of inequalities? ( 'l--7,1:\. (-!-)'1..- l L.._ 1 - ~ + 1 _L I (show work and plug in each ordered pair} "', 1 J... _ -x + y:::;; 1-3 / 'i ~ \ True -!. 6. I \w<.. \~ -true. ~t" ~~ i ~ i 5. ~so\ unci\ @ (1/2,1} c. (3,4} (1,1} ' d. (5/2, 6} 2. Shade the graph to represent all of the solutions to the following system: -1 +J ~I 0!:: r l'?ue. (Graph each by ha.nd. Then show a test point for each function then shade appropriately) y:::;; xz + 2x- 8 fh.tc&>o\r.. Te~A--{9 1 c) o b.c} +2(c)~ 0!:. -<6.. 'Fa.\be. ;-s~e ou\<b \cit, L\1\e. le'bl-- (-10 rc) 0 )3(.-\0) 0=~'2.-t-zx-~ o= cx.-ta.t)('l'-2) y > 3x +c ~\l\-\- )(-; -1-J X-= 2. l...~(lt..": _, M\N l-\ J-e;") (""'\)7. +2.( -i)-~ 'I \(\+ ( 0,-cr;') 0 '> T~~ )co\'\ru\ ~U1Clll\\ w()j\{ix)o\d-.,\()::e/\o~ hnt 1.. 3. Find all solutions to x 2-6x + 7 < 2. For the parabola, you must include the x-intercepts, line of symmetry, mc~:x/min point, y-intercept. Sketch a graph to show how the left side compares to the right side. Then 'display the solutions using a number line and interval not~tion. Vo.foJ:>o\0.. ln+ej5c:g-hoy\ 'X"2:. L9. '>G +I = D )( 2 - l.t '>< t-t = 2. y_-; ut~zor-i) '1. L_ t;.x +- 5= o - 2. (X- ~)(x-l) =- D -x~ ~~~ x~ 5 x== I '2- '(~ 4.t4 J XX J. 5't Lineo.f~ymrn X= t;_ ::3 ry\ \.N ( '?> 1-2)??- {..('3}t l 9-1~+1-2 )I 1n+ (o ;r)
. L \'N- ~ 'YrutJ.ro\o_ - 4. Find all solutions to 2- x ~x 2-2x. For the para bola,-you must include the x~intercepts, line of symmetry; max/min point, y-intercept; Sketch a graph to show how the left side compares to the right side. Then display the solutions using a number line and intervalnotation. '"Po.-r<:k.'oo\CL L \A e. I n Wbe.cti o (\ b-=-x-z--2.:'j. '/ = l-~ 2-Y. =- y, 2 - b D=~lx-z..> 1::.-x..,.z.. -t)( -rx "'--=--o '1.'=2 ~-=x-z-x. L\A-c. X-=-1 M\~ (, ri) - z_ -2 0=-x.,.-x-z. D = (x-z:y)(+l) X-=- 2. X-=--t 5. Express the following in interval notation (-L_ )S] a -4 b -;;. -1 2 d x>2.orx~-5 d\clw D.._ - = lif\e.+d \:'\e\-v 6. Express the following solutions on a number line. C -4 <X< 3 (- 0o; -5]u ( 2, L>OJ a. ( 2,5) b. (-oo, 6] U [ 8, oo) C. [ 3, oo) 2-5 \. ( ) ). L6:1 $\ fl9 6 v"7y9.
7. The height in yards h{x) of a football kick x yards away from the kicker is shown on the graph. For each question: write a sentence to describe the mathematical sentence AND provide a solution in interval notation. ';;;' 20 +- -- ~.. ~- ;p ----~--~''' ;: ~-..,---- -- "';! ~ 15 -!- -'-7'-----'-- = =- ~ io ---1----...,-~-----'----->t-,-~,!:>1 "ffi ::X: 5 -!- [ ~ - -:!)+-~------~~----~~~ (l 10 20 s:o 40 50 60.70 Horizontal Dtstance (in yani$) a) Estimate the solution to h(x) = 10 x~~ \J:: I D Wha:J Ib ~ ~-fo.!l C{ 1 W \-\.Q.':-~ fbo+k:cj I i $ l 'D y(acc\0,f"\~ O:,r ~ c) Estimate the sol,ution to h(x) > 10 6.1 ~~CL.;~t~i't OD,s-11\t ne I~ )l.t o-t -fht bo...\~ o.'oo"vt. \O'fcl1. ( ~~'5&) d) Estimate the solution to h(x) ~ 10 A+ whaj d/ sf o.n ce s is -J1u be).. I/ a+ or be low ID yd m #uair? [.oj<6]u[si 1 ~.,qsj or L-,.. 1 'ii"] U LSi' 10')..b)"Estimate the solution to h(10} \,.D~cd-\..D-\k hjla~ 0~ ~ -tta..y'oo)..\ ~"-sl..\\. ~ 'oa..\\.,<f> \(.) '10~ ~on'-~t.iilir: e) Write an inequality whose solution will indicate how far the ball must be from the kicker, in order for it to be more than 20 yards off of the ground. h(x) '>20 ( 22) '-f2) 8. Factor the trinomials and solve. Show all work! a) y = x 2 + 8x + 7 b) f(x) = x 2 - x- 20 c) g(x) = x 2-10x + 21 0-:: (X-r/)(x:+i) 0 -= (><- '5 )(_x+4) 0 ~ (K -I') lx.-.3) )(.:: -7 ")('::--1 x.,? x~-j.f Yr--7 X-=-3
8. cont.'d Factor the trinomials and solve. Show a II work! d) y= 2x 2 - Sx -12 ~lj p \ 1\~ e) m(x)= 3/ + 8x- 3 Q::: z~z-~.(t3x-l2. Q;; 3x:z---+qX,-)><--3 _ -.:::. u uc-t-t)...- W-t;) v 3 x (-x: +3)- \(>'+~ =- (2-:K+- 3'J (X-ti) = (?,x. -bcj<.t--3) X~ -3(z_ x~ji )C==' '13 x::::-3 9. Given that f(x) = 2x -1 and g(x) = x 2-7. Find:.. 2.. a)(f + g)(x)= X t 2 X- '6. 2. 2x -1 t- X -7 h) (f- g)(2)= l.q c) g(3) = Z ~[2-)-5(2-)::-- 3 - (-~):::: u 32:. ry -;. q -I :;. 2_.r (?) ~- 2(2)-l ::: 3 d) (fg)(x)~ L.x-1)()(1;. I)= 2.x 3 - /I./X- X~+-~- ~\~ - e)(~ 2xz.-\ s ~ (x2--2jj -J Z.x- 1'-l-1 "' f)(g~ -ij.f(\)- 2.(\)-1 kf1. :2. :~v 9 (, )=1-7 = -~o 10. Perform the following operations with complex numbers. a) i= r-\ b) (3 + 7i) + (4- Si) - l -J-L_i._, c) (3 + 7i) - ( 4 - Si) ~ - } + J ll d) (3+7i)x(4-5i).: Lf] +-/3i tvil- IZ -15L+2WC-35i 2 J 2 _,_ t3l- 35c-') I L r 13[ 1-3 5 ---- --
------~~~----- - 11. Fi ~he x and y intercepts of the following functions. X.(\0~- a) F(x) = x 2 + 5x + 6 yf\ 6.:=- {X.+2)('K-t3) k=--z )(;-3 Xrv'lt (:-zl o) (-3,D) b) F(x) = x 3-6x 2 0::: x_z. ( x.-lp) )(o::. 0 X= lo c) F(x) = /-25. b ::: ()(-t-5)(x.-5) x-~ -? x.~ c:/ 12. Find the vertex of the following functions. On a) you must put the quadratic into parabolic form by completing the square, on b).use either method and show all work. a) f(x) = x 2-4x + 6. ~('X '2-4y_t-{),-!/._ t-le -. 2- ~NVJJJ\\G 2 ~ ' - 0 D ={x-2) + 2.. -- -ver-t-ex M\tJ (1-1 2) G!Jf ()\1\ fh b) f(x) = -3i + 24x -4 _ fi(st~t{:{t~v ~ -3(~7:-sx+lw;-r..'i -t-j v.... o{\ ~ ~ fv'i11 ~. 2- -"~','"' ~ \211~iC -. -3 (x-:4y +-44 ~~AX {'fji-f'-i) 13. Describe the right and left hand behavior of the following function. JustifY your answer using the appropriate mathematical terminology. (degree, turning pts, leading coefficient, etc.) (is the left increasing or decreasing?) no graph necessary. a)f(x)=7x8x'}c2 ~ lj~+ \f\cjejc\si~ fl.:k- 4.J!l7x t 2 ---t-1 ~'"\j\\1- eke\@ Ylj rj DnllU 3 -. v~~~ (li5ca tw e._ b)f(x)~-~5(~:6.~..v.p-- ~}. f ' f ' r_ {)IL ""', k f o.t1d ryn Jf)C rea!3lr:j L po-s --- ------'---~~~ :,
14. Sketch a graph of the following functions. Be sure to clearly show the locations of the zeros and pay attention to the multiplicities and their behaviors. (YOU MAY HAVE TO FACTOR FIRST) 2- -(9 3 s z_.? ~ I'D l a) y = x 4-3x 3-10i z. ( 'Z...) o=x )<.. -3x.-lo = X.2.(x.-5)l'K+2) ~M 0 L 5 I Tf~ -2. 1 tye~. t.f LC..3 b) y =2x 3 - SOx '1.. =l.x( X - Z..'S) ::: 2x (_x -6)()(.t-6j K_~ r~o t.edd\1\j - - 5 _ I toef.i~ - S I ~i.snot~roo+ c) y = {x- 1) 3 (x ~ 4}(x+3) 2 {x + 6) I Lf -3 -( 'D3 TPZ u...+- 3 I z d) y = -2{x +4/ (x + 6) 3 (x-5) 2 (x - 1) 3 peqrt,-pq LCne~ve. t>l Wtv LC...- 15. Find the vertical asymptotes, horizontal asymptote, x-intercept, y-intercept, and domain of the following functions. Use correct notation for all answers. HA VA )<. \r'l +- )' i ntzx-3 a) F{x) = x+l x~... I (3f2JO) (oi-3) J=-2 v~ (-f'p)- i)u( -L) lti) X b) F(x) = - 2 - x -9 (Xr~){x-3) Y=-o ~.. ~ -~--------
y:; '/ ~. ( t1= 1 o) ( b :#-) 15. cont.'d Find the vertical asymptotes, horizontal asymptote, x-intercept, y-mt~rcept, and domain of the following functions. Use correct notation for all answers. VA ~ \V\ :r yifl+ HA c) F(x) x 2-2x-8 (x.-j-1)(~~ X~ -1. X-=4- (Op) (o,o) d) F(x) = x+ 3 x 2 +3 )\z...-3::;0 (l'c(~()..\ 5o\ oova 16. Combine and simplifythe following rational expressions. a)4x+12 q ('if 3). xz-g [X+-3)0<-3) l-1'..1'(/z,) bf2x-!:>l-' + x+3 (Jt.) x ()c.-2) x-2 (7\) L'~-~~) Sx-6 x+1 (J<.t'1) c) x+4~) x-6 6.+-... u 5x'2--3o~-.'It' t3/t, - ()l.r.i-f)lx'-u.) X 2.;-t.ty.. r{.h Lf ( -.;. t-t-ij(>g-lt).a-1"1'\ 4x x 2-9 ro.ctv:a d)-.- l~f\l:u 2x--6 5x 2 3x?- l;x+ ID X('i-:Z) e)x 2-5x+4 + 6x-6 x 2-4x,... 3x lflipt.~ 5x 2-3Ltxt-3l - ( X 2 f5xr'-t) (_ xr4)lx-l4 ') if xz.. - 31ld:..-:.3:...;:2..=-..._ &t t-j)b:.-u) ':..(_ x_-_'-~...:.x_ 1( -.:...rj _:::..3~x =- ~ = ~ X (x.-lo CD 0'-f) l9x 2. 17. Find the inverse of each function. Show all work and use correct notation. a. f(x) = 3x ~ 4 b. g(x) =..Jx + 7 c. f(x) = (2x- 4) 3 d. Find f 1 {40) when f(x) = 3x 2 + 7 X so t.to is y () fl f(- Cx) tix~"y rx ~d."'-y ~x -t- LJ ~~'~ '2 t -/x ); -~ -+--4 d..
VJY'i.fe C6, /os ~Rf) CJrJCAl1)t ot 0Q6e_ 18. Solve each exponential equation. Round to the thousandth. I);:J a. 3x=120 I o J 3 120 :: X. b. 4x-S = 615. /OJ Lf ~If= X~ 5 ~~)t. I o Cj ~ "'-~ J-/.?'5~ C. 5(2x) = 255 5 5 2)( = 51 } Dc_j1.. 5/=-X _!_c::/s 1. = " 'ID j 2- e. 3 2 x-? +10 = 98 -ro -,o J_o5 t; \5.:_ -x-? l DC() 4 X % q. u. 3;;{ L./. u 3;t~ X-'? -rs _ t-s d. 7(3} 5 x- 4 = 468 7 7 3 \o5:; ( ~ )~ 5x-L/ X%).5(15 '5x~~- LM - I l 0 ~ ( '4\s/6/1) ~ 2 ')(-{ {.g2x= -13 3.~ 138 X~ 5.?3~..JN (\ t-jf IOCJ 3..;J:j f. 6 2 x- 7 = -20 +I -+7 lo3 ~ ~ ~d.x-7 Jo3&-13 =2X' 3 f) ~ ~ #Jfl.l-+i ()D solu on I D~ 3 wci-k CJ...D e1<-):bf\.h)'h'oj 19. Solve each logarithmic equation. Round to the thousandth. a. log 3 (2x+1)=8 3'6 = 2x+l cps~\:: 2-)(+I -1 _, 05uo:: d.~ ~l~ ;:--}( b. log 2 x +4 = 15 -'t -'-j 1CXjz.X=li d. 21og 6 (5x-4) = 10 2- z IO<j /..Q ( 5x-'-1) = 5 & 5 ==- 5x.-'-/ 11 7Lt: '3x-l/ " '" 1--V