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Accelerated Integrated Geometry 6.1 Properties of Exponents * Definition of Exponents: ais areal number, “is a natural number a" zaxaxax,..xa,n times, a°=1 1 ! a a" as 4 a"= * Properties of Exponents: Let aand bbe nonzero real numbers. Let mand be integers. Product: (a)" (a)"=a™ m-n Quotient: =a ssa ab)" = a"b' _ b" e Examples: Simplify and write with positive exponents. | Re Power of a Power: — 2 Power of a Product: Power of a Quotient: ol 1 xtxt@z xo 2. Ze zo 3. Gy ee (Product) Caucliert) 2,18 Add (Power of a Power) 5. Satb*(-2a°b*)f= 90° L?(-2") (a8)'(b")= Fo? b* (4 a?b®) = pee) _ *eetee IE 4. 3x?y? (-2x3y* )e-6x8y" y 6. = Codec) Product al ower ofa Product) a ~ -27b%'> ; abe eb ee abt -21b S42 Fale = teat = eo? 21¢8 (a-')* (b?)* a*b pe ath” TAI -t 4 (Gueliert a] oer ofa Guation)) pis Sgt gt Pe. puto po _ i a’ ben 7 “eat” Accelerated Integrated Geometry 6.2 Operations with Functions «Definitions: Function - relationship between two variables (x,y) where every x has one and only one y Domain - all possible x-values Range - all possible y-values « €xamples: Tell whether each relation is a function. If it is, state the domain and range. 1. {(0.1),(2.6).(3,-2)} 2. {(-4,5),(-1,5).(3,15)} 3, {(3,12),(3,t0),(17,1)} Yes Yes No Yrnain: 6, 2,3 Domine: 4, -1, 3 Range: I, @,-2 Range: 0.5,1.5,5 Vertical Line Test: If a vertical line intersects a given graph at no more than one point, then the graph is a function. + Examples: Use the vertical line test to tell whether each graph is a function. If it is a function, then state the domain and range. oo 4. \ * 5. = 6. an eae NE 4 , noe H+} Yes — Vv. Domai: (-os, oo) All real #'s -k Rouge: -leyed or El, 1] No Yes Deon 32243 or 3,3) e Function Notation: a fancy way of writing “y" . cy way 9 Y Rong@i-2£ ye 2 ow £2,2] y=2x+5 —————_-» f(x)=2x+5 "fof x"=2x+5 Uses any variable: g(x),h(x), j(x),b(x).... e Examples: Evaluate each function for the indicated value. 7. f(x) =2x?-7 for x=9 8. gx)=5x' forx=-1 $(4)=2(a)-7 ged aC - f(a)=2(61)-7 gC 30) F(a)= 155 gi: $ x°- 6x49 Accelerated Integrated Geometry 6.3 Composition of Functions « Composition of Functions: uses substitution... (feg)(x) = F(900) “f composed of 9’ * Examples: Perform the indicated operation and find the domain and each resulting composition. Let f(x)=2x? and g(x)=x". 1. (Feg)(x)= la) | 3; nein 2. (ge DEQ Ge) Doman Qc?)= 2¢3= where x0 Geto ee Le where 40 Let f(x)=x*-1, g(x)=3x+1, h(x) = ex ,and j(x) = oom Then, find the domain. 3. (g°f)(%)= g(FCx)) 4. (Fea))=#(g(2)) 3(EI) +12 pe 3+1=3x7-2 (xa)? 1e Cone (30- I)-l= Dornainz All real #° 8. (Fea)(2)=F( (2)) Ga2+ Bxt3ntl-l=4x74+ Ox Domain = All real 4's 6 (Fegeh)(x)= £(g(h))) £(3@)+!) = Fi)= (e248 £(3(3z) +i) Sloe) f@= x! No domain 7. (Fe j)(~) -3\2% 2)% Gay.) = — -|= X-3x-3x44 16 -l= Domaine Altrec! #'s — -6xt9 1 g2-Gx-7 IG IG 167 16 “tes + pal’ 4x4 2x4 2a41-|> 44244 Vornain= All reel #°s 8. (he j)(&) £(22). 26 26-9) (a3) “TR = G Accelerated Integrated Geometry 6.4 Inverses of Functions ¢ Inverse Function: interchanges the input and output values of the original relation The domain of the inverse is the range of the original relation. The range of the inverse is the domain of the original relation. ¢ Notation: The inverse of function f is denoted by f". One-to-One Function: For a function to have an inverse function f", the original function must be one-to-one, which means that no two elements in the domain of f correspond to the same element in the range of f. If the function f is increasing or decreasing in its entire domain, f is one-to-one, The Horizontal Line Test can also be performed to determine if f is one-to-one, ¢ Examples: Determine whether each function is one-to-one graphically. Then, state the domain and range. Corwerse) Festive, se is go up 3 3 $ ; b AO) = 3x48 4 Tae 2, 96x) =(x-3)' - SxX+9 aT izka f ~ 4? ae L fS 7X I Yj || Dornaine All veal 4's : ne Range = All veal #'s Not one-to-one fonction bne-do- one dunction Yornaive= All real H's + Examples: Find the inverse of each relation. Ranges y2-Z 3. {(1,2),(2,4),(3,6).(4,8)} 4. {(-2,4),(3,-4).(-8,-5)} Inverse = £02,0), (4,2), 6,3), ( ans Twerse = $42), C 4,3), ¢5-8)% * PROVE GRAPHICALLY: If two equations are inverses, then their graphs reflect about the line y=X. el je i could be inverse equations if inverse is a function. x , 2 11, ye onsy> ye3x3y 3 ax One-to-one function yeXx -HOWEVER... The inverse in this example is not an inverse function. ‘Sis not on inverse functions, because it does not pass the verdical line 4est * Examples: Sketch the graph of each function and its inverse. State whether the 12, y=x*-3, xz0oney3 atsey y2O ye a) xt3 * be pep a Po vertical line test indwidually mpt both ! One-to-one function 13. y=x?-3,x 1.00 + 1.002 2.00 or 100% + 100% = 200% 1 hr. = 33(2)*= 132 3 br. = 33(2)°= 2,112 b) 225 bacteria that triple every hour Trifial amount = 225 Mulliplier > |.004+1.00+1.00= 3,00- — + bw.= 225(3)*= 675 3 hr. 2 225(8)°= 6,075 Accelerated Integrated Geometry 6.7 Exponential Functions ¢ Exponential Functions: f(x)=b”*, where bis the base and xis the exponent « Growth and Decay: b>1: exponential growth O as X > -00 Right End: f(x) > as X > +00 e Analyzing Graphs of Exponential Functions: Graph of y=a*, a>1l Graph of y=a™*, a>1 Domain: All veal nombers R) All real numbers (®) Range: Lo, too | [o, +00] Intercept: Y = 4 Y= | Increasing/ Decreasing: Tnereasi nO, Decveasi n5 Asymptote: Y= O y - 6 Accelerated Integrated Geometry 6.8 Transformations of Exponential Functions « Transformations: parent function > y = b* y =ab" +k a: Vertical Stretch/Compression c: Horizontal Stretch/Compression whole #: stretch bya whole #: compression by 1 c fraction: compression by a fraction: stretch by 1 c negative: reflection about x-axis negative: reflection about y-axis k: Vertical Translation move in exact direction of sign h: Horizontal Translation move opposite direction of sign « Examples: State the transformations from each of the given parent functions, (x), to the given relative function, g(x). f(x) =5* #00 -(5) 1 (3 2. 4-3)} -{x-2) g(x) = (5) +7 g(x) =-5 - ‘Reflection over y-axis — Reflection over x-axis 1 ts Uorizontal stretch by 4 . ‘ 22 Right on the z-axis by 2 +3: Lefton the x-axis by 3 f(x) = (2) . g(x) = 2(2)" +4 @ i Horizontal compression by t +43 Up the y-axis by 4 +712 Up the yraxis by 7 f(x) =2*? -4 * g(x) = ~3(2") +1 LR: Horizontal shift left 8 #: Vertical strelch by 3 — = Pebleation across the L-OXKIS $1: Vertical shift up | « Examples: Graph each parent equation with its given relative equation. State the domain, range, and the coordinates of the y-intercept for the relative equation. _ x42 2 x: 2 x41 5. y=3 > y=2(3) -1 6. y-(§) sy=3l3 ~3 i TOTALS Perot Alt reol nuerbers [AL [ Domain: (0,99) forge: [1,408] Porae: (3,0) } Yeiedercept: y= 17 yr evopt:(0,-!) i }__| r= —<> L yA > * Application of Exponential Functions: Compound Interest Formula a annual interest rate it nt Total investment <—— A(t) =P] 1+ F time in years = (amount over Principal (initial) # of imes compounded e Examples: 7. Find the amount of a $100 investment after 10 years at 5% interest compounded annually and quarterly. prnually * AQ)=$00(14 222)" = £162.84 Guarderly : Ale)» ®100(4+ 288)" = 9 164, 36 8. Find the final amount of a $500 investment after 8 years at 7% interest compounded monthly and daily. Morhly : Ple)= S500 (4+ 202)" = 6 873.71 8 Daily : ACA)= S50) (1+ 282 007)" - $973 47 + w nn tua wou " Would you prefer a monthly allowance of $100 or to start with $1 and double it each day, i.e. on day 1 you would receive $1, day 2 - $2, day 3 - $4, etc? Give arguments for your choice, I prefer 4p staré with Slaldewle i+ each day, because of you devia + each day hen det amount will get bigger compare 40 9100 0 mowlh, State the transformations from each of the given parent functions, f(x), to the given relative function, g(x). f(x) =5" #00 -(3) 1 b) ox) =") 1)*?) vorizowhal shift left 3 a=(5) 7 Horizontal treteh by 4 +7=\Nerdical shift up 1 2-2 = Horizontal shidt righ 2 Nertical stretcln by 5 ~ = Reflection over y- ois Reflection over %- axis #0 =2(§) , fega2 4 1 /(<8) 6x =-+(29)) 41 o(X) =2 3) +4 960) =-3(29) + +4zvertical ht up 4 . dal Ae Left 10 al shi &= Horizontal compression by Yo Weir cst up S° 2=\erhiral stretch by 2 Fe Vertical compression by 5 = Reflection ower x-axis nt Compound Interest formula: A(t) = P(t + 4) Find the amount of a $100 investment after 10 years at 5% interest compounded annually and quarterly. annvally . jarderhy P= 100 A(ie) = 100 (1+ 228)! '0 Alo) = j00 (142.08 )*"° +? 10 R= 0.05 $162.89 5164.36 Find the final amount of a $500 investment after 8 years at 7% interest compounded monthly and daily. month |: P= 500 A(s)= 560 (14222 28 A(8)=500 (ingzye" tc ae 9873.4 $375, 29 N= |2,365 
