PRECALC ALG & TRIG
PRECALC ALG & TRIG MAC 1147
Popular in Course
Popular in Calculus and Pre Calculus
verified elite notetaker
This 31 page Class Notes was uploaded by Marquise Graham on Saturday September 19, 2015. The Class Notes belongs to MAC 1147 at University of Florida taught by Larissa Williamson in Fall. Since its upload, it has received 13 views. For similar materials see /class/207048/mac-1147-university-of-florida in Calculus and Pre Calculus at University of Florida.
Reviews for PRECALC ALG & TRIG
Report this Material
What is Karma?
Karma is the currency of StudySoup.
Date Created: 09/19/15
L3 Polynomial Division Synthetic Division Rational Expressions Long Division 15 5426 l 5 426 2 Check Dividend QuotientDivisor Remainder Dividing by a monomial 4x5 6x3 x2 2x2 2x2 4x5 6x3 x2 Dividing two polynomials with more than one term 1 Write terms in both polynomials in descending order according to degree 2 Insert missing terms in both polynomials with a 0 coef cient 3 Use Long Division algorithm The remainder is a polynomial Whose degree is less than the degree of the divisor Example Perform the division 3x4 6x2 12x 4 3x2 2 3x2 0x 2 3x4 0x3 6x2 12x 4 Synthetic Division Synthetic division is used when a polynomial is divided by a rstdegree binomial of the form x k ax2 bxc x k k a b c Coef cients of Dividend Diagonal pattern Multiply by k Vertical pattern Add terms Example Use synthetic division to nd the quotient and remainder 2x4 3x2 5xl x 1 Example Verify that x 3 is a factor of x3 x2 10x 6 Rational Expressions The domain of an expression in one variable is the set of all real numbers for which the expression is de ned A rational expression is a ratio of two polynomials The domain of a rational expression is the set of all real numbers which do not make the denominator equal to zero Reducing Rational Expressions 31 ifb 0c 0 be I Example Find the domain and reduce the expression to lowest terms 2 1 x 5x6 3x9 Domain 62 25 2 S x Domain Multiplication and division 252 ifb 0d 0 b d bd i ifb 0c 0d 0 b d b c Example Perform the indicated operations and simplify Give restrictions on the variables 3 y 8 4y 2 2y3 y2 5y 6 x2 y2 ltx y2 Addition and subtraction In order to add subtract rational expressions we use the Least Common Multiple LCM of the denominators To Find the LCM of the Denominators p A Factor polynomials that are in the denominators N The LCM is a product of all different factors of the denominators numbers variables expressions each raised to the largest power that appears on that factor Example Add or subtract as indicated Give all restrictions on the variables x x3 Note Be aware of the case when the denominators are additive inverses of each other Example Perform the indicated operations 2 5 x 2 2 x 3x 5x x2 x Z Mixed Quotients A mixed quotient eomplex aetion is a quotient of rational expressions Simplifying a Complex Fraction Method 1 Multiply both numerator and denominator by the LCM of the denominators of all simple fractions and simplify Method 2 Perform the indicated additions andor subtractions in the numerator and denominator and then divide Example Simplify the complex fraction mixed quotient 1 x xl 2x l x L x xl Example Write the fraction as a sum of two or more expressions x3 5x2 4 x2 Example Perform the indicated operations and simplify i x h2 x2 h L15 Polynomial Functions and Inequalities The palynamiulfunclian is a function of the form x anxquot aHx39 alx do where an any a1 a0 are real numbers and n 2 0 is an integer Degree 0 fx a0 a0 i 0 Degree 1 fxa1xa0 5110 Degree 2 fx azxz a1x a0 all i 0 Features of the Polynomial Function 1 The domain is the set of all real numbers 2 The graph has no holes gaps or jumps 7 a polynomial function is continuous 3 The graph has no sharp comers or cusps 7 a polynomial function is smooth Power Function A pawerfunclian of degree n is a function of the orm f x xquot where n gt 0 is an integer Properties of the Power Functions 1 y xquot where n is even 1 Symmetry End Behavior 2 y xquot where n is add Symmetry End Behavior 2 The greater the value of n the steeper graph of y xquot when x lt 71 or x gt1 and the atter graph of y xquot when 71lt x lt1 155 Graphing Polynomials A number 0 is called a zero ofa polynomial fx if fc 0 x c is a zero ofa polynomial fx if and only if x c is a factor off The number of times the factor x 0 occurs in the polynomial is called the multiplicit of the zero 0 Example Find all zeros and their multiplicities in parentheses fx 2x2 x 43 x 2x2 1 The F 39 39 Theorem of Algebra Every polynomial of degree 1 or more has at least one complex zero Number of Zeros Theorem A polynomial of degree n has at most n distinct zeros Note The real zeros of a function are the xintercepts 156 Multiplicities of Zeros and Graph Sketching Example Find the intervals on which f x gt 0 and fx lt 0 iffx x32x l Solution Zeros of fx x 3 of multiplicity 2 7 even x 1 of multiplicity l 7 odd Intervals 0073 x 3 391 x 1 194 x 32 x 1 o fxx32xel 0 0 Observe 1 If the multiplicity of a zero is even the corresponding to the zero factor does not change sign passing through it If the multiplicity is E the factor changes sign 2 fx changes sign at a zero if and only ifthe factor corresponding to this zero changes its sign 157 Therefore End Behavior 1 does not change sign touches the xaXis when passing through a zero of an even multiplicity Example Describe the end behavior of the graph of fxx3 x2 9x9 Hint factor out x3 2 f x changes sign and consider lxl gt so crosses the xaXis when passing through a zero of an odd multiplicity Turning Points Turning points of the graph are points of local maxima or local minima A polynomial of degree n has at most n l turning For large values of lx the polynomial pOInlS fx anxquot an71xquot 1a1xa0 a 0 n behaves as its leading term y anxquot Also as x gt 00 the polynomial f x has the sign of its leading coef cient atquot 158 159 Example The graph of a polynomial f x is given Tell a if the degree of fx is even or odd b if the leading coefficient is positive or negative c What is the smallest possible degree of fx Example Match each function to the graph a fx 2x4 2x2 b gx x3 x2 2x c 11x l4x5 34c3 x 234 lniual L Jgilumwjmmqmm gt Analyzing the Graph of a Polvnomial Function p A 5 6 7 Check for symmetry Find all real zeros xintercepts and their multiplicities Find the yintercept Give the number of turning points Locate the turning points if possible Analyze the end behavior Plot all intercepts Using the end behavior and multiplicities of the zeros determine the signs of the polynomial on each interval With xintercepts as endpoints 8 9 Plot a few additional points if necessary Draw the graph Example Draw the graph of fx 73x4 12x2 Solving Polynomial Inequalities 1 5 e Move over all terms of the inequality to the le hand side with a 0 on the right side and simplify in order to obtain one of the inequalities fXS0 fx20 fxlt0 fxgt0 where f x is a polynomial Find all real zeros of f x and their multiplicities Put the zeros on the number line and label them as ifthe inequality is non strict fx g 0 or fx 2 0 the zeros will be included in the answer otherwise label them as 0 Determine the signs of f x on each interval with the zeros as endpoints To do that First nd the sign of the leading eoe ieient and set it on the rightmost interval the end behavior when x gt 00 Then starting from the rightmost interval and moving to the le on the number line alternate sign when passing through a zero of odd multiplicig39 leave the same sign when passing through a zero of even multiplicig Fquot gt1 Select intervals with the desired sign according to the inequality in Step 1 Give closed intervals in your answer if the zeros are included39 otherwise give open intervals Use the symbol U to join two or more intervals 16 Example Solve the inequalities 2 Example Find the domain ofthe functionfx 2 V9 x2 2x 3x lt 5 Vertical Leap Record Guinness Book of World Records reports that German shepherds can make vertical leaps of over 10 feet when 2x 51 x23 x3 Z 0 scaling walls Ifthe distance s in feet off the ground is given by the equation s l6t2 24t 1 for how many seconds is the dog more than 9 feet off the ground L21 Applications of Exponential Function and Logarithms Logistic Models Simple Interest Formula If a principal of P dollars is invested for a period of t years at a per annum interest rate R expressed as a decimal the interest earned is I P R t The interest is called the simple interest Compound interest is the interest paid on the principal and previously earned interest Compound Interest Formula The amount A after t years due to a principal P invested at an annual interest rate r compounded n times per year is m A P 1 5 n Note The more frequently the interest rate is compounded the larger n the larger is the amount of A Question Is it true that A gt 00 as n gt 00 241 Example Suppose that a principal P l 00 is invested at an annual interest rate r 1 100 compounded n times per year a Find the future value A after t 1 year b What value does A approach when n gt oo In general I l gt00 limP l L P6quot n Continuous Compounding The amount A after t years due to a principal P invested at an annual interest rate r compounded continuously is A P6quot 242 Example If 5000 is deposited in an account at an interest rate 6 how much will be in the account after 10 years if a compounded quarterly b compounded continuously 243 Example How long will it take for 500 to grow to 6000 at an interest rate of 10 per annum if interest is compounded a daily b continuously 244 Exponential Growth and Decay The exponential model is used when the quantity changes with time proportionally to the amount or number present At 2 A06 where A0 AO is the original amount or number and k i O is a constant Uninhibited Growth of Population Nt Noek k gt0 Uninhibited Radioactive Decay At Aoek k lt O 245 Example A sample culture contains 500 bacteria when first measured and 1000 bacteria when measured 72 minutes later a Determine a formula for the number of bacteria N t at any time t hours after the original measurement b What is the number of bacteria at the end of 3 hours c How long does it take for the number to increase to 5000 246 The halflife is the time it takes for a half of a given amount to decay Example Find the halflife of iodine131 used in the diagnosis of the thyroid gland if it decays according to the function AOe 00866t where t is in days 247 Example Paint from the LascauX caves of France contains 15 of the normal amount of carbon 14 Estimate the age of the paintings if the halflife of carbon 14 is 5730 years 248 Applications of Logaritth The pH of a chemical solution is given by the formula pH log H1 where H1 is the concentration of hydrogen ions in moles per liter pH 70 water pH lt 7 acidic solution pH gt 7 alkaline solution Example Find the pH of the solution for which H 16 x102 limes 249 Richter scale Magnitude of an Earthquake An earthquake whose seismographic reading measures x millimeters at a distance of 100 km from the epicenter has the magnitude M x given by Mx iogl x0 where x0 10 3 mm is the reading of a zerolevel earthquake at distance of 100 km from its epicenter Example Determine the magnitude of an earthquake in Japan in July 1993 whose seismographic reading measured 63095734 mm at 100 km from the epicenter 250 Logistic Functions Logistic growth model can be used when the value of the dependent variable is limited as the time elapses Logistic Growth Model In a logistic growth model the dependent variable P after time t obeys the equation Pt C l ate l where a b and C are constants with c gt O and b gt O Note 1ft gt 00 then 6 gt O and Pt gt c The number 6 is called the carrying capacity Example P 473 problem 22 The logistic growth model 090 P z 0 135e 339 relates the proportion of new personal computers sold at Best Buy that have Intel s latest coprocessor t month after it has been introduced 251 a What proportion of new personal computers sold at Best Buy will have Intel s latest coprocessor when it was first introduced that is at t 0 b Determine the maximum proportion of new personal computers sold at Best Buy that will have Intel s latest coprocessor carrying capacity c When will 075 75 of new personal computers sold at Best Buy have Intel s latest coprocessor L19 Exponential and Logarithmic Functions Example For f x cal a gt 0 a i1 0 0 evaluate Statement If a and x are real numbers with a gt 0 and fx 1 a 1 then y ax is a uniquely de ned real number fx Example For fx ax b a i 0 evaluate fx1fx Laws of Exponents Ifs t a and b are real numbers with a gt 0 b gt 0 then X Comparing Exponential and Linear Models as at ast at asit as t art a0 1 a For an exponential model fx eax a gt 0 a 31 ab a by if a a7 i Z e i 0 for unit increases in the input the output changes I b5 ax a by a factor of a base fxlfxa The exponential function with the base a is a For a linear model f x ax b a i 0 for unit function of the form increases in the input the output changes by an additive a fx ax slope Whereagt0anda l fx1fxa The domain of f is the set of all real numbers Example Find the appropriate model for each of the problem Wages of an Assembly Line Worker A manufacturer pays its assembly line workers 1150 per hour In addition workers receive a piecework rate of 075 per unit produced Write the equation for the hourly wages Win terms of the number of units x produced per hour Size of a Population Under ideal conditions a certain bacteria population is known to double every hour Suppose that there are initially 100 bacteria Write the equation for the size of the population N after t hours Graphing Exponential Functions fx 2 fx 2quot Note Use these graphs as templates for graphing fx a with a gtl on the left and O lt a ltl on the right p Graphing 39 Functions Using T Example Use the graph of fx 2 to graph the following Give the domain and range 300 2H hm 2 2 kx 72 Properties of the Function fx ax a gt 0 a 1 1 Domain Range 2 Points 4 01 and La are on the graph a 3 The line y 0 is the horizontal asymptote ifagt1then ax 0 as x oo if0ltalt1then a 0 as x oo 4 fx a is increasing if a gt1 decreasing if 0 lt a lt1 5 fx ax is a onetoone function Recall f is onetoone means that lt3 uv Note An equivalent form of the property 5 is u v a 0 IV This property is useful for solving exponential equations Example Solve the equations GT 11 9x 2317x The base 2 Graphing y ex Consider the expression 1 for n 1 23 n The values for some natural n are listed below 71 n1 11 2 n 1 n 7110 l 2593742 n n100 11 27048l3 n n10000 11 27l8l45 n nl000000 11 27l8280 n nl000000000 11 2718281 n Note 11 271828 as mam n We denote 271828 e and write lim1 e Hm n Logarithmic Function The exponential function y a a gt 0 a 1 is onet0 one function Therefore its inverse is also a function which is called the logarithmic function and denoted y loga x According to the properm of the inverses logaxzy ayzx Also Domainloga x Rangeax 0oo Range loga x Domaina oooo The Cancellation Rules for logarithms and exponents loga ax x for all real x Itng a 1 forxgt0 We can use the cancellation rules when converting from logarithmic form to the equivalent exponential form or vice versa 0 When converting from the logarithmic form loga x y to the exponential form x ay we compose the exponential function with the base a to both sides and simplify 0 When converting from the exponential form ax y to the logarithmic form x loga y we compose logarithmic function with the base a to both sides and simplify Example Write in equivalent exponential form logz z Al loge2x l4 Example Write in equivalent logarithmic form Graphing Logarithmic Functions 1 4 71 3 81 4 4 Graph the functions y 2 and y log2 x Use the graph of y log2 x as a template for graphing ylogaxagtl Example Solve the equations log33x l 2 Use the graph of y logl x Ni N as a template for graphing ylogax 0ltaltl Knowing the properties of f x ax and relations between the inverse functions we can obtain the properties of the logarithmic function f 71x log x f x ax f 1x10ga x Domain oooo Domain Range 0oo Range y 0 is HA Points 11a 01 Points 1a are on the graph a are on the graph Increasing if a gt139 Increasing if decreasing if 0 lt a lt1 decreasing if