COLLEGE ALGEBRA MAC 1105
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This 10 page Class Notes was uploaded by Nathen Fadel on Thursday September 17, 2015. The Class Notes belongs to MAC 1105 at Florida State University taught by Staff in Fall. Since its upload, it has received 169 views. For similar materials see /class/205606/mac-1105-florida-state-university in Calculus and Pre Calculus at Florida State University.
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Date Created: 09/17/15
Study Tips Just as a great pianist must spend many hours practicing to develop his or her talent7 and ath letes must devote many long hours to practice in order to have good skills7 so it is with mathematics To become good at math7 one must practice7 practice7 practice Make a schedule Choose a time each day to do your math homework Do not have a radio or television on while you are working STAY FOCUSED ON YOUR WORK Allocate about two hours each day for math homework and study Many students schedule this time in the Math Center to take advantage of the resources there Find your mistakes and correct them Rework any problems that you miss reworking the problem until you can do it without a mistake Keep Read do and understand the examples in the book and in your notes The best way to learn mathematics is to do mathematics To do well you must get out pencil and paper and rewrite both the book and your notes You must wrestle with every step in every example Fill in the details that are often omitted from the text If you do not understand a step7 you must keep returning to it until you do You must work the homework with the same diligence Memorize de nitions and basic identities Attend all class sessions on time You cannot know exactly what goes on in class unless you are there Missing class and then expecting to nd out what went on from someone else is not the same as being there yourself Read the book It is best to read the section that will be covered in class beforehand Reading in advance7 even if you do not understand everything you read7 is still better than going to class with no idea of what will be discussed Work problems every day and check your answers The key to success in mathematics is working problems The more problems you work7 the better you will become at working them The answers to the odd numbered problems are given in the back of the book After you complete each problem7 be sure to compare it to the answer in the book If you make a mistake7 nd out what it is and correct it Do it on your own Having someone else this includes the solutions manual show you how to work a problem is not the same as working the problem yourself It is okay to get help when you are stuck As a matter of fact7 it is a great idea Just be sure that7 ultimately7 you can correctly do the problem on your own Review every day After you have nished the problems I assigned7 spend at least another 15 minutes reworking problems from previous homework assignments both the easy and difficult ones This will keep you from forgetting the material7 and get you better prepared for the next test Don7t expect to understand every new topic the rst time Sometimes you will understand everything you are doing7 and sometimes you won t That s just the way things are in mathematics Expecting to understand each new topic the rst time you see it can lead to disappointment and frustration The process of understanding mathematics takes time It requires that you read the book7 work problems7 and get your questions answered Spend as much time as it takes for you to master the material No set formula exists for the exact amount of time you need to spend on mathematics to master it You will nd out as you go along what is or isn t enough time for you If you end up spending two or more hours on each section in order to master the material7 then that s how much time it takes trying to get by with less will not work MAC 1105 COURSE OBJECTIVES Updated for F 09 Sec Obj Course Objectives Optional text homework problems are also listed Primary homework problems are the online problems available at the russell eGrade site See Syllabus pg 2 for the web address A5 Hwk 111 odd Pre 1 Preliminary Objective P1 Understand the relationship between a 7 b and b 7 a Pre 2 Preliminary Objective P2 Factor quadratic trinomials 1 Rational Expressions a Equivalent Rational Expressions Class Examples b Multiply or divide rational expressions simplify Sec A5 Example 2 c Add or subtract rational expressions simplify Sec A5 Examples 376 d Simplify mixed quotient Sec A5 Example 7 A6 Hwk 37 39 41 43 47 57 61 63 65 2 Understand the meaning of rational exponents simplify numbers raised to rational exponents Sec A6 ample 7 3 Special Factoring Techniques a Factor by grouping refer to your class notes may also refer to Sec A3 Example 3f b Factor and simplify an expression containing rational exponents Sec A6 Example 10 c Factor and simplify an expression containing rational exponents and a common binomial factor Class Examples CN Hwk Sec 13 151 every other odd Sec 15 115 and 3153 every other odd 4 Solving Equations a Solve linear equations Refer to your class notes may also refer to Sec A1 Examples 12 b Solve quadratic equations including Quadratic Formula Refer to your class notes may also refer to Sec 13 Algebraic solution of Example 6 Sec A2 Example 3 c Solve rational equations Class Examples d Solve higher order equations Understand the existence of real number roots refer to your class notes may also refer to Sec A1 Example 13a e Solve equations that contain even or oddiroot radicals Refer to your class notes may also refer to Sec 13 Algebraic Solutions of Example 10 5 Inequalities a Solve linear inequality and express the solution in interval notation Refer to your class notes may also refer to Sec15 Examples 7 8 b Express the solution to inequalities in interval notation Refer to your class notes may also refer to Sec15 Example 1 c Express the solution to inequalities in interval notation and understanding the terms or and and Refer to your class notes 11 Hwk 1 21 23 31 33 49 55 57 6 Rectangular Coordinate System a Understand plotting points on the Rectangular Coordinate Sytem Sec 11 Figure 2 3 b Recall and use the distance formula Sec 11 Example 2 c Recall and use the midpoint formula Sec 11 Example 5 12 Hwk 3 7 9 For 11 19 25 27 sketch the graph by hand by making a table of values nd any intercepts 31 33 37 41 47 49 51 53 54 57 59 61 63 For 65 67 69 71 73 find the intercepts test for symmetry you do not need to graph 7 General Graphing Principles a Understand what it means for a point ab to be on the graph of an equation Sec 12 Example 12910 b Identify intercepts from a graph or from an equation Sec 12 Example 45 c Symme Determine symmetry with respect to the xiaxis yiaxis or origin from a graph Sec 12 Figure 27 Given a point on a graph give the coordinates of a point that must also be on the graph if the graph is symmetric with respect to the xiaxis yiaxis or origin Sec 12 Example 7 A 39 determine if the graph of an equation has any symmetry Sec 12 Example 8 16 Hwk 175 odd 85 87 89 8 Linear Equations a Calculate and interpret slope Sec 16 Example 1 b Graph lines by hand by obtaining the x7 and y intercepts or any two points Sec 16 Example 2 3 c Identify the slope and yiintercept from the equation of a line Sec 16 Example 7 d Write the equation of a horizontal or vertical line Sec 16 Example 3 5 d Write the equation of a line given two points on the line or given a point and the slope Sec 16 Example 4 e Write the equation for a linear relationship described in an applications problem Class Examples f Write the equation of a line that goes through a given point that is parallel or perpendicular to a given line Sec 16 Example 9 10 11 17 Hwk For 5 7 9 11 Just write the standard form 15 17 You will not be asked to complete the square and obtain the general form of the equation ofa circle 9 Identify the center and radius and graph a circle when given the equation in standard centeriradius form Sec 17 Examples 12 21 Hwk 1 3 5 9 For 13 15 17 19 add g nd f3a 2132 all 33 35omit c 3745 odd 46 47 4962 all 67 69 10 unctions a Identify the graph of a function determine whether a relation represents a function Sec 21 Examples 127 b Find value of a function Sec 21 Example 4 c Find the domain and range of a function from a graph Sec 21 Example 8 c Find the domain of a function from the equation of the function Sec 21 Example 6 d Obtain information from and about the graph of a function Sec 21 Examples 89 23 Hwk 17 odd 9 11 13 15 19 25 31 33 37 39 4149 odd 55 63 65 71 11 Properties of Functions a From a graph identify intervals where a function is increasing decreasing or constant Sec 23 Example 3 b From a graph identify local maximums or local minimums and where they occur Sec 23 Figure 24 c Find the average rate of change of a function Sec 23 Example 2 d Find the slope of the secant line containing xfx and x hfx 11 on the graph of a function y fx Sec 23 e Determine from a graph or from an equation whether a function is even or odd Sec 23 Example 56 12 Recognize the graph equation and properties of any of the basic functions in the Library of Functions except Greatestilnteger Sec 23 13 Functions defined Piecewise a Evaluate a function defined piecewise Sec 23 Example 7 b Graph a function defined piecewise Sec 23 Example 7 24 Hwk 123 odd 2943 odd 59 61 63 14 Graphing with Re ections Compressions Stretching Translations a Identify reflections about the x7 or yiaxis graph a function reflected about either axis Understand the affect of a reflection about a coordinate axis on the coordinates of a point on a graph or on the domain or range of the function Sec 24 Figure 46 b Identify compressing or stretching factors from an equation graph a function with these Understand the affect of a compressing or stretching factor on the coordinates of a point on a graph or on the domain or range of the function Sec 24 Example 3 c Identify vertical or horizontal translations from an equation graph a function with these Sec 24 Example 12 25 Hwk 19 odd 1327 odd 31 33 37 47 49 51 15 Form the sum difference product or quotient of two functions evaluate give the domain of the new function Sec 25 Examp e 16 Function Composition a Form the composite of two functions evaluate a composite function Sec 25 Examples 2 4 b Find the domain of a composite function Sec 25 Example 3 26 Hwk 1a 3a 17 Construct and analyze functions and math models Sec 26 Examples 175 31 Hwk 17 odd 1321 odd 25 29 35 37 39 41 43 43 49 53 57 59 61 63 65 67 71 73 75 abc 18 Quadratic Functions a Given a quadratic function in the form y axZ bx c find the vertex all intercepts and sketch the graph by hand Sec 31 Examples 1 7 5 b Given a quadratic function in the form y ax 7 h2 k find the vertex all intercepts and sketch the graph by hand Apply graphing translations from Sec 24 c Obtain the quadratic function needed to solve an applications problem find the maximum or minimum value of a quadratic function Sec 31 Example 7710 32 Hwk Figure 19 and 20 19 Power Functions a Graph a power function by hand give domain and range and identify intervals where increasing or decreasing Sec 32 Figure 19 20 38 Hwk Solve algebraically 1 3 9 11 17 25 27 33 39 41 45 47 49 53 20 Polynomial and Rational Inequalities a Solve a polynomial inequality algebraically Sec 38 Example 1 2 b Solve a rational inequality algebraically Sec 38 Example 3 c Given a rational inequality find the rational inequality needed to identify partitioning values Class Examples 41 Hwk 1 3 5 9 11 15 17 19 For 21 2527 verify and graph For 29 just verify 3335 37 39 41 For 47 49 53 just nd the inverse 21 Inverse Functions a Determine whether a function is oneitoione by looking at a graph or set of ordered pairs Sec 41 Example 2 b Given the graph of a oneitoione function draw the graph of the inverse function Sec 41 Example 4 c Use composition to determine if two functions are inverses Sec 41 Example 6 d Given an equation of a function find an equation of the inverse function f39l Sec 41 Example 6 7 42 Hwk For 1118 omit D H G do 11 12 1517 For 1924 omit F do 19 2124 25 27 37 39 41 and 45 1931 22 Exponential Functions a Given an exponential function give the domain range intervals where increasing or decreasing find intercepts when possible sketch the graph by hand Sec 42 Example 2 3 b Given an exponential function with a translation give the domain range intervals where increasing or decreasing find intercepts when possible sketch the graph by hand Sec 42 Example 45 c Use a calculator to evaluate r t39 A 39 39 c uding A A quot 39 problems Sec 42 Example 1 d Solve t39 equations by obtaining the same base Sec 45 Example 4 43 Hwk 121 every other odd 2549 odd For 5360 omit D G H do 53 54 5759 For 6166 omit E F do 61 63 65 66 6773 odd 23 Logarithmic Functions a Evaluate logarithmic functions exactly Identify when logarithmic functions are defined and when not defined Sec 43 Example 4 b Given a lo arithmic function give the domain range intervals where increasing or decreasing find intercepts when possible sketch the graph by hand Sec 43 Figure 25 c Given a logarithmic function with a translation give the domain range intervals where increasing or decreasing find intercepts when possible sketch the graph by hand Sec 43 Example 6 7 d Find the domain of a lo arithmic function Sec 43 Example 5 44 Hwk 131 odd 3541 24 Properties of Logarithms a Understand when and how to apply basic logarithm properties Sec 44 Examples 12 b Understand the inverse function 39 391 between F t39 and quotL 39 functions Simplify expressions using this relationship Sec 44 Example 2 c Write a logarithmic expression as a sum or difference of logarithms Sec 44 Example 3 4 5 Write a quotL 39 expression as a single logarithm Sec 44 Example 6 45 Hwk 111 odd 1523 odd 3139 odd 4553 odd 25 Solve Exponential Equations a Solve exponential equations algebraically Sec 45 Example 789 b Solve exponential equations algebraically when base is e or 10 Class Examples 26 Solve Logarithmic Equations a Solve logarithmic equations algebraically Sec 45 Example 2 b Solve logarithmic equations algebraically using the definition of logarithms Sec 45 Example 1 3 27 Solve other kinds of equations involving exponential functions Class Examples 28 Solve other kinds of equations involving quotL 39 unctions Class Examples 46 Hwk 1 1329 31 33 37 29 Compound Interest a Future Value or Present Value with quarterly or monthly compounding Sec 46 Examples 13 4 5 b Future Value or Present Value with continuous compounding Sec 46 Examples 3 4 5 c Determine time required to double or triple an amount of money Sec 46 Example 7 101 Hwk 3 11 15 17 21 25 30 Solve algebraically 2 linear equations in 2 unknowns interpret the solution graphically Sec 101 Example 479 107 Hwk 1 5 11 31 Solve 39 a system of nonlinear equations in two unknowns Sec 107 Examples 12 108 Hwk 1 3 9 11 13 21 23 32 Linear Inequalities a Graph a linear inequality Sec 108 Examples 1 3 b Graph a system of linear inequalities Sec 108 Examples 4679 VmmeL Ewm E E H a o nm 2m mm o gmii m damenm fu omn m 2 010101 3 9n 3 rammYAiiwn 19207 02644 NXimGMm Emir m dimCWM Sn rOmmeuqx wn gzmm 6 om5215 mxmammmu z 3 mxmvzaz pf fad zlmx ozg r Nympmmmmaz 40 95nv nwa 1rwns gt 0091302 0 p 655933 Wm i 4i mmcwmomamAcm ammfhdwlc 5 999212 mx 2 16 46c 9 9mm lm dmmvzhjdl 3 pm v1 3 F Sowij 3 mome mgp um u 5 i V VOmPX XP 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u fisf a fluva mm pz fi d 3sz amp EOE 5 I x 23 s 323 uagz N mmzsgwp k awgu aTE 3do WEL 92239 muz m d fm Umb1 kvmxm 2 392 1 3 21 9 quotRAW Van3 35 9 wt 8 9 32 e IN UX My A Au NTJznTQOBNrE 0290a 259 Nm X W MLNY A rmeovi K 033 A E G033 Amg yaw mOd gnaw J 5w x NW X A l d m W wtmvxa A nuXMurmf yaw 6 Xm Aux Ax a VSud quotll A A m A 1me A x9 u x f 1 fan luxi xm A A AL A Al x A A w Ram i 3 z 23km 5 3 w 3 3 E K oev 3 0wa umqomfmv or ww wkro mw 88 g 2 do 0232 30 o 288gt 3 an A 9 Co m3 w J 463w mwgoz HE Pf macaw door 133 9 6 mm m thaw wwad dogn aooa gbzgfa 8325 3r Guam y A A A Aw 7mapvm do 0 A kioL 91 rq wme A undH F 2 u CG 35 5 7 15 g A 33 A2 A mummu3f33fu nni lm x f grim A wif Bagm xf E 1 g 6 WmmztxmydoE MACHOE Extra Logs and Exponents Problems 6263 Rewrite the following as a exponential or logarithm 4 log e I Solve each of the fbllowing equations for a exactly 39 1 20 22 24 113 I 39 1 2 60599 l 39 implify using the logarithm properities 39 logp210g Solve using the logarithm properities Hint Same as above just solve after simplify 39 39 1 log1003 5 i 39 H3 1 2 391new2139 3 quot 3 iog77 r u Using the properties of logarithams discussed in class and in homework Write the expressions as a sum and or difference of logarithms Express powers as factors Assume all variables are positive 1 logo 3y2z3 336197 10gb 4 W i 3 log 9333 2 a 33 41n 5 1nrTE4 39 A