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Unformatted text preview: ' "7 ‘L' ' 22
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Ex: Let f(x) = x3 — 5x2 mx+ 5. First find the x and y intercepts. W
ehav10r of f[x)W Weiwcmlty by ﬁgurmg out Intervals whe x > , orf(x)<0. Note: Factors of f[x) correspond to the roots (zeroes) of the graph of the function. Find HAL XriM‘(S> 0W0Q \jJI/‘Jr
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J {1(a) A» J11 Hg mm»? 70:?) WW)“; ' amily of functions that we are going to learn about are the
rational functions. T e_ are of the form: = Ii , where, (x) and g(x) are both polynomials :
x ,2,_._ ,_' Functions have breaks (discontinuous) unl'
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smooth and continuous __.___—— i 1
EX: f(x)=—, x¢0
x The vertical dotted line where the function
above is‘called the vertical asymptote. " E “" l To figure out the vertical asymptote for rational function of the form )2: —}all we have to do is to ﬁnd the x—values that will make g(x)= 0. These x—values $75]le make y go to infinity (positively or negatively larger and larger] which 1n turn make the
vertical asymptotes. +. 2
Ex: Find‘ the vertical asymptotes of the rational function y = x m 5.3.5.2 3 , _ . x 3"g(x)
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W >534) )> 56"} $3 QAWEW Similarly Horizontal Asymptotes are deﬁned as x gets positively or negatively larger
Mr [approaches inﬁnity] what y— Vaiue dries the rational function , x + 2 _f(x) «m
y — — —— approach ,
x— 3 g(x)
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Ex: Find the horizontal asymptotes of the rational function y = x = ﬂ ‘
x —— 3 g(x) , 1 Observe: Any term of the form 7 —> 0 as X gets positively or negatively larger and
x larger. % "+3: Cox/gm N‘ H'Av ===> if: _ ( .
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m___ X“\ Ex: Sketch the graph oi y = i— l Specify the intercepts, asymptotes and whether . x _ 4 F...___,__7 u , I. W
(“WxJahe graph ever crosses on al Asymptote. Include all the above information your sketch. ' Note: A rational function usually consists of several branches separated by 7 _ asymptotes. An asymptote is a line such that the distance between the line and
l curve approaches zero as it gets larger and larger. The last important thing to remember while graphing rational functions 15 can the .
function ever cross the Horizontal Asymptote? WWW;W ~WWWWu lave/wow? JW‘WC ‘3—— ”Xi‘2 ' Omssivxg; . w” ”V3 ' sq) was 1?! f (g ‘ 2'. “+2 W ' 7 e \ WA/ ‘3,  Q ”Xx—3) equaljﬁw “Use the example above to find at what values of x if a y the rational function' IS 7 crossing the Horizontal Asymptote. W 2 _
9m W20 95[ W6 N0 Orbss‘wﬁ‘ ‘ I. ; _ Graph the functio y— —— x+ 2 $32 by labeling all th Important Note: A function can never Cross the f'ertlcal” symptote
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EX: Sketch the graph of y =. x2 :4“: . Specify the int cepts, asymptotes and
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whether the graph ever crosses the Horizontal Asymptote Include all the above
information in your sketch. . Note: Pay speciai attention to the domain._ 3 ‘—— /____._____.— "Does ”63%“
«ax/or 62/333391» type? Slant Asymptote will Show up in the homework but not in your exams: W _. 2
1 x“. Then Ex: Show that the line y = —x is a slant asymptote for the graph y =
. x sketch the graph with the asymptotes and intercepts. 31W be {SM/3 Weds Wu it W
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Asymptotes (Vertical, Horizontal, orSlant) Ex: Graph the function y = by ﬁnding out the x—int(s), y—int [5), All Chapter 5: Exponential and Logarithmic Functions Exponential Functions [5.1) W )(WWM Definition: Let b denote a positive consta ther than 1. The exponential function 7 with base b is defined by the equationi b > 0, b #1 New 1 TM hm 1) 25m
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{3 [i ) _ _ la # te 23: x2 y: «f; are no r. examples of exponential functions. T ase has to
be constant' In order to quahf as one. ﬁnal Range: ‘
CO )
i
Difference between the two ra hs:
g p M m “We? RJ‘M 33$ QAC’EIA' 3W? agmém‘ﬁ WINS Now on I will not plot points to refer to a standard exponential function graph.
Instead the shape as above will be drawn for y = b", b > 0, b #1 NOW SkEtCh : y 2 __2x and COmpare it to the graph of 12:.) I
 V7  ..,_:E“"E, % h Range; (ﬂ 09) ﬂ) w Difference between the two graphs: .
t. "x \ q _ x .L
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 plotting points by tl‘gl 'iag‘nug amngj 1' 2X EX: Sketch =4" on the same set ofaxes. Ex: Sketch 3222", y .3" How does the change in the base value affect the shape of the general look of an exgnentialgraph? Vm L9 be COWS gym
111 11111 MW ”WVWW Properties of Graphs of Exponential Functions: Let for): b", b > 0 b #1. Then the
graph of f(X]: 1] Is continuous for all real numbers 1 “\f
2) Haswxs._ [{4“
(01 I) 3) Passes through the point @a 4) Lies above the x axis which" is a hor ontal asymptote either as ‘9) ~— 0 x—> co or x —> —oo but not both 5] Increase as x increases if b>1, decreases as x increases"  1
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6) ﬁx) 15 one to one, t at IS intersects any horizontal l .1 e at most once.
g_______._.. Solve the following exponential equations:
Ex: Solve for t: (1?” z 9"6 I W3 A? WQ +k1 (bet/L0. __ .. +0 be He 3W 4% 1m
Life :: (”ELYg @390 . ’ E :“E £3 RQW‘. 7‘ ‘5
Ex: Solve for x: 372‘” = 9"” _ 2 E ,
C58 )2“: [39‘ 7M "qt :4 ~12,
3(11 14) “—1 ll“) t Z '5
Cm: 4% 2w '=~> 71:45:?— ==7 f—‘v Properties ofreal numberexponents for 17", b> 0, b :21 ' lo 7 ' W l0 4 '
example: 331'? example: 32 2 .
example: 33 =%/3—2 _ j— Wu
4) If xis irrational, pproximat . b" with b’" where r :‘x example 3‘5 z 31'2 ' ___?" m Remember:All the properties related to exponents will hold true for 14 ...
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 Spring '08
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