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# COLLEGE ALGEBRA MATH 1021

LSU

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This 22 page Class Notes was uploaded by Maverick Koelpin on Tuesday October 13, 2015. The Class Notes belongs to MATH 1021 at Louisiana State University taught by D. Kopcso in Fall. Since its upload, it has received 11 views. For similar materials see /class/222624/math-1021-louisiana-state-university in Mathematics (M) at Louisiana State University.

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Date Created: 10/13/15

Section 46 Rational Functions and Their Graphs Defmition Rational Function A rational function is a function of the form f x where g and h are polynomial functions such that hx 0 W Objective 1 Finding the Domain and Intercepts of Rational Functions Rational functions are de ned for all values of x except those for which the denominator hx is equal to zero If f x has a y intercept it can be found by evaluating f 0provided that f 0is defined If f x has any x intercepts they can be found by solving the equation gx 0 provided that g and h do not share a common factor 4615 For the given rational function determine the following a the domain b the yintercept if any c the xintercepts Objective 2 Identifying Vertical Asymptotes Defmition Vertical Asymptote The vertical line x a is a vertical asymptote of a function y f x if at least one of the following occurs Y 11 y 196 Ay Ky 4 fx I 3 PM 3 Egg a ixa i x ixa yfx fx gtooasx gta fx gt ooasx gtafx gtooasx gta fx gt ooasx gta A rational function of the form f x Where gx and hx have no common factors will have a x vertical asymptote at x a if ha 0 It is essential to cancel any common factors before locating the vertical asymptotes If there is an xintercept near the vertical asymptote it is essential to choose a test value that is between the xintercept and the vertical asymptote 46710 Find all vertical asymptotes and create a rough sketch of the graph near each asymptote The vertical asymptote is x Objective 3 Identifying Horizontal Asymptotes De nition Horizontal Asymptote A horizontal line y H is a horizontal asymptote of a function f if the values of f x approach some xed number H as the values of xapproach 00 or oo 11 1 3 1 2 3 4 5 x y f x The line y 71 is ahorizontal asymptote The line y 3 is ahorizontal asymptote The line y 2 is ahorizontal asymptote because the values offx approach 1 because the values offx approach 3 because the values offx approach 2 asx approaches 00 asx approaches iOO asx approaches iOO Properties of Horizontal Asymptotes of Rational Functions 0 Although a rational function can have many vertical asymptotes it can have at most one horizontal asymptote The graph of a rational function will never intersect a vertical asymptote but may intersect a horizontal asymptote A rational function f x that is written in lowest terms all common factors of the x numerator and denominator have been cancelled will have a horizontal asymptote whenever the degree of hx is greater than or equal to the degree of gx Finding Horizontal Asymptotes of a Rational Function n nil n72 gx anx an71x an72x alxa0 Let foo h b quotb m 1b m 2b b aquot 0 bm 0 x mx milx mizx 1x 0 where f is written in lowest terms 71 is the degree of g and m is the degree of h o If m gt n then y 0is the horizontal asymptote o If m n then the horizontal asymptote is y 21 quot the ratio of the leading coefficients m o If m lt n then there are no horizontal asymptotes 461320 Find the equation of all horizontal asymptotes if any of the rational function There is a horizontal asymptote at y Objective 4 Using Transformations to Sketch the Graphs of Rational Functions The graphs of fx L and fx i2 x x Properties of the graph of f x l Properties of the graph of f x i2 x x Domain oo0U0oo Domain oo0U0oo Range oo0U0oo Range 000 No intercepts No intercepts Vertical Asymptote x 0 Vertical Asymptote x 0 Horizontal Asymptote y 0 Horizontal Asymptote y 0 Odd function f x f x Even function f x f x The graph is symmetric about the origin The graph is symmetric about the yaXis 462126 2930 Use transformations of f x l or f x i2 to sketch the rational function Label all intercepts and nd x x the equations of all asymptotes Objective 7 Sketching Rational Functions Steps for Graphing Rational Functions of the Form f x x 1 Find the domain 2 If gx and hx have common factors cancel all common factors determining the x coordinates of any removable discontinuities and rewrite fin lowest terms 3 Check for symmetry If f x f x then the graph of f x is odd and thus symmetric about the origin If f x f x then the graph of f x is even and thus symmetric about the yaXis 4 Find the y intercept by evaluating f 0 5 Find the xintercepts by nding the zeros of the numerator off being careful to use the new numerator if a common factor has been removed 6 Find the vertical asymptotes by nding the zeros of the denominator of being careful to use the new denominator if a common factor has been removed Use test values to determine the behavior of the graph on each side of the vertical asymptotes 7 Determine if the graph has any horizontal asymptotes 8 Plot points choosing values of x between each intercept and choosing values of x on either side of the all vertical asymptotes 9 Complete the sketch 464748 Follow the ninestep graphing strategy to sketch the graph of the rational function Section 52b The Natural Exponential Function Objective 4 Solving Applications ofti1e Natural Exponential Function Continuous Compound Interest Formula Continuous compound interest can be calculated using the ula A Pen where A Total amount after t years P Princip al 7 Interest rate per year t Number ofyears 5221 wiuuieiebeinan L of8yearsifm1 39 quot p quot quotata7 that is compounded continuously Exponential Growth A model that describes the population P after a certain time t is P PM 071 Pt PM 1 where PU P0 is the initial population and k gt 0is a constant called the relative growth rate Note kmay be given as a percent In 1975 a quot quot 39 a 39 39 39 39 forest for the rst time In 2007 the rabbit population had grown to 6454 The relative growth rate for this rabbit species is 22 a How many rabbits did misc 4 39 L forest in 1975 b How many rabbits can be expected in the year 2021 Section 55 Exponential and Logarithmic Equations De nition of the Logarithmic Function For xgt0bgt0andb l if ylogbx then xbyandifxbythenylogbx The Logarithm Property of Equality For bgt0 bil ugt0andvgt0 if logbu logbvthen u v and if uvthen logbu logbv Properties of Logarithms For bgt0 1321 ugt0andvgt0 10gb uv 10gb u 10gb v Product Rule for Logarithms log b E log b u log b v Quotient Rule for Logaritth v 10gb ur r 10gb u Power Rule for Logaritth Change of Base Formula For any positive base b at land for any positive real number u then where a is any positive number such that a 72 1 Note that the preferred choices for a are 10 and e Objective 1 Solving Exponential Equations Solving Exponential Equations If the equation can be written in the form b bv relating the bases then solve the equation 14 v If the equation can be written in the form b c where c is a constant not equal to b 1 Rewrite the equation in logarithmic form using the De nition of a Logarithmic Function 2 Solve for the given variable and use the Change of Base Formula to evaluate If the equation cannot be written in the form b bv or b c 1 Use the Logarithm Property of Equality to take the log of both sides base 10 or base e 2 Use the Power Rule of Logarithms to bring down any exponents 3 Solve for the given variable 553111 and 7 in this order Solve the exponential equation Round to four decimal places where necessary for an approximate answer Objective 2 Solving Logarithmic Equations Solving Logarithmic Equations If the equation can be written in the form log n log v then solve the equationu v If the equation cannot be written in the form log n log v 1 Determine the domain of the variable 2 Use Properties of Logarithms to combine all logarithms and write as a single logarithm if needed 3 Use the De nition of a Logarithmic Function to rewrite the equation in exponential form 4 Solve for the given variable 5 Check for any extraneous solutions Verify that each solution is in the domain of the variable When solving logarithmic equations it is important to always verify the solutions The process of solving logarithmic equations often produces extraneous solutions 551619 22 and 26 Solve the logarithmic equation Section 51b Exponential Functions Objective 4 Solving Applications of Exponential Functions Periodic Compound Interest Formula Periodic compound interest can be calculated using the formula V nt A P1 where n A Total amount after t years P Principal original investment r Interest rate per year 71 Number of times interest is compounded per year t Number of years 5135 How much money will you have in 10 years if you invest 13000 at a 38 annual rate of interest compounded quarterly How much will you have if it is compounded monthly Sec on 525 The Natural Exponen al Func on Obj een39ve 1 Understanding the Chal zL39IEl iSlil s nl39 the Natural Exp nnmrial Funeu nn lquotas h approaches mflmty L 05 roundedto 6 l l The funetroh fx e ls ealled the natural expnnential rnneu39nn l L f0 x h negraphmheha yex tun erpurrenralrmlhn r 2 r 1 00 00 271 10000000 100 000 000 Characteristics nrthe Natural Expnnenu39al Funetinn Th NmrlF nuwh The domam of x exrs awe and the range ls 0w The graph of fx 2X rhterseets theyeaxls at 01 2X am as x aw 2X gt0 as r area The lme y 0 ls ahonzontal asymptote The funetroh fx e rs 011371070113 521and4 Objective 2 Sketching the Graphs of Natural Exponential Functions fxex 528 Use the graph of f x ex and transformations to sketch the exponential functions Determine the domain and range Also determine the yintercept and nd the equation of the horizontal asymptote Objective 3 Solving Natural Exponential Equations by Relating the Bases The Method of Relating the Bases for Solving Exponential Equations Ifan exponential equation can be written in the form b bv then u v 52 14 Solve the exponential equation using the method of relating the bases by rst rewriting the equation in the form b bv Section 19b Rational Inequalities Objective 2 Solving Rational Inequalities A rational inequality can be solved using a technique similar to the one used to solve a polynomial inequality except that the boundary points are found by setting both the polynomial in the numerator and the denominator equal to zero Since division by zero is never permitted the boundary points that are found by setting the polynomial in the denominator equal to zero must always be represented by an open circle on the number line You cannot multiply both sides of the inequality by a variable term to clear the fraction This is because we do not know if that term is negative or positive therefore we do not know whether or not we would need to reverse the direction of the inequality 191 1 14 Solve the inequality Express the answer using interval notation Section 51a Exponential Functions Objective 1 Understanding the Characteristics of Exponential Functions De nition of an Exponential Function An exponential function is a function of the form f x bx where x is any real number and b gt 0 such that b at 1 The constant b is called the base of the exponential function Characteristics of Exponential Functions For b gt 0 b at 1 the exponential function with base b is de ned by fx bx The domain of fx bx is oooo and the range is 000 The graph of fx bx has one ofthe following two shapes fxbquotbgt1 fxbquot0ltblt1 The graph intersects the y axis at 01 The graph intersects the y axis at 01 The line y 0 is a horizontal asymptote The line y 0 is a horizontal asymptote 513 Sketch the graph of the exponential function x 7 516 Determine the correct exponential function of the form f x bquot whose graph is given x Dhi eeuve 2 Sketching 11m Graphs ur Einmnential Functinns Using Transfm mztlnn y 5 5 Thegaphuffx3xrl yeix A mu utmiwmnysum 3 3 y 3 t the gsphafy 3 dawn uneuntt 1 L3 ii lhegnphuf yZ 2 in m u M x 2 Use the graph of y 3 Also detenntne the ydntercept and nd the equation othe honzonta1 asymptote Obi eedve 3 Snlving Expnnmtia Eqnzdnns by Relating hE Base The function fx bxls onertorone because the graph off passes the honzontal hne test t L L t L bu v t t The Methud nfRelzting the Base at Snlving Expnnential Equz nns Vthenuv If an exponential equatton ean be wntten tn the form A 5 1 3 26 and 30 Solve the exponenttal equation usmg the method ortelattng the basesquot by rstrewnnng the equation tn the m b bquot Section 54 Properties of Logarithms Objective 1 Using the Product Rule Quotient Rule and Power Rule for Logarithms Properties of Logarithms If b gt 0 b t 1 u and v represent positive numbers and r is any real number then 10gb uv 10gb u 10gb v Product Rule for Logarithms 10gb 210gb ui 10gb v Quotient Rule for Logarithms v 10gb ur rlogb u Power Rule for Logarithms W39arning 10gb u v is NOT equivalent to 10gb u 10gb v 10gb u 7 v is NOT equivalent to 10gb u flog V 1 M is NOT equivalent to 10gb u 7 10gb v 10gb v 10gb u r is NOT equivalent to r 10gb u 544 5 7 or 9 Use the properties of logarithms to expand the expression Wherever possible evaluate the expression Objective 2 Expanding and Condensing Logarithmic Expressions 5413 Use the properties of logarithms to expand the expression Wherever possible evaluate the expression 5427 Use the properties of logarithms to rewrite the expression as a single logarithm Wherever possible evaluate the expression Objective 3 Solving Logarithmic Equations Using the Logaritlun Property of Equality The Logarithm Property of Equality Ifa logarithmic equation can be written in the form logb u logb v then u v Furthermore if u v then logb u logb v 5434 Use the properties of logarithms and the logarithm property of equality to solve the equation Objective 4 Using the Change of Base Formula Change of Base Formula For any positive base b 72 land for any positive real number u then where a is any positive number such that a 7i 1 5440 Use the change of base formula and a calculator to approximate the following expressions Do not round until the nal answer Then round to four decimal places as needed 5447 Solve the equation and simplify the answer Section 53 Logarithmic Functions Objective 1 Understanding the Definition of a Logarithmic Function Every exponential function of the form f x bx where b gt 0 and b at l is onetoone and thus has an inverse function The graph of fx bx 1 gt1 and its inverse To nd the equation of f71 Step 1 Change fx toy y bx Step 2 Interchange x and y x by Step 3 Solve fory Before we can solve for y we must introduce the following definition De nition of the Logarithmic Function For x gt 0 b gt Oand b at 1 the logarithmic function with base b is de ned by ylogbx ifandonlyifxby Step 3 Solve fory x by can be written as y logb x Step 4 Changey to f1x f 1x logb x 531 Write the exponential equation as an equation involving a logarithm 539 Write the logarithmic equation as an exponential equation Objective 2 Evaluating Logarithmic Expressions The expression log b x is the exponent to which b must be raised to in order to get x 5313 Evaluate the logarithm without the use of a calculator Objective 3 Understanding the Properties of Logaritluns General Properties of Logaritth For bgt0andb l l logbbl and 2 logbl0 Cancellation Properties of Exponentials and Logaritth For bgt0andb l 1 blogbxx and 2 logb bx x 5321 and 25 Use the properties of logarithms to evaluate the expression without the use of a calculator Objective 4 Using the Conunon and Natural Logarithms Defmition of the Common Logarithmic Function For x gt 0 the common logarithmic function is de ned by ylogx ifand onlyifx10y Defmition of the Natural Logarithmic Function For x gt 0 the natural logarithmic function is de ned by ylnx ifandonlyifxey 5327 28 Write the exponential equation as an equation involving a common logarithm or a natural logarithm Dhiective 5 Undarnandjngthe Characteristits nl39Lnganthmm Functinns aracteristics nangariLhmic Functinns For A gt 0 A 1 the logarithm funenon thh base A 15 de ned by ylogb x The domam of x logb x15 use and the range 15 70000 The graph of x logb xhas one ofthe followmg two shapes fUVWSM 51 Ln x 1Ag x fxlogbx Agt1 fxlogbx 0ltA lt1 5 3 34 35 Wnte the logarithm equahon as an exponential equahon Obi mm 5 Sketching the Graphs ul39Lngaridunic Funr nns Using Tmsrmnn atinns 5 3 46 of any vertical asymptotes Dhj eccive 7 Finding r11 e Dnm ain at L ngarithmic Funcrinns Le x 10 ggx then the domam offcan be found by solvmgthe inequality W gt 0 5 3 53 Fmdthe domam ofthe logarithme funehon Section 56 Applications of Exponential and Logarithmic Functions Objective 1 Solving Compound Interest Applications 562 How long in years and months will it take for an investment to double if it is invested at 8 compounded monthly 563 How long will it take for an investment to triple if it is compounded continuously at 4 two decimal places Objective 2 Exponential Growth and Decay Applications 566 The population of Adamsville grew from 6000 to 13000 in 8 years Assuming uninhibited exponential growth what is the expected population in an additional 3 years Exponential Decay A model that describes the exponential decay ofa population quantity or amountA after a certain time t is Au AM where AG A0is the initial quantity and k lt 0 is a constant called the relative decay constant Note kis sometimes given as a percent HalLLire n 39 39 39 L halflire 39 39 39 39 a given quantity ofthat element to decay to half ofits original mass 5510 A stream L A 4 me rean1z ofthe mynal 5512 A A certain element A650yearold m superhero s nemesis The halflife ofthe element is known to be 250 years a How many grams ofthe element are still containedin the stolen rock b A P

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