PRECALCULUS ALGEBRA MAC 1140
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This 10 page Class Notes was uploaded by Marquise Graham on Saturday September 19, 2015. The Class Notes belongs to MAC 1140 at University of Florida taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/207050/mac-1140-university-of-florida in Calculus and Pre Calculus at University of Florida.
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Date Created: 09/19/15
SUPPLEMENTAL 1 Some helpful formulas The following formulas should be memorized Business Notation C cost R revenue P pro t x of units demand p price per unit C variable cost fixed cost R xp P R C Geomet Notation b base h height l length w width r radius d diameter P perimeter C circumference V volumeA area S surface area Triangle A bh Pythagorean Theorem a2 b2 c2 for a right triangle Rectangle P 21 w A W Circle C 2727 7rd A In2 Rectangular box V lwh S 21w 21h 2hw The following geometry formulas are useful but they do not have to be memorized in this course Right circular cylinder V rzh S 27rrh 27239quot2 Right circular cone V rzh Sphere V 7273 S 47239quot2 SUPPLEMENTAL 2 Factoring a trinomial Step 1 In order to factor a trinomial first check if it is a perfect square trinomial If so factor it according to the perfect square formula Step 2 If it is not a perfect square trinomial try to use FOIL in reverse Step 3 If this method does not work or you are experiencing trouble do the following If the trinomial is in the form ax2 bx c evaluate the discriminant D b2 4610 If D is a perfect square for example 36 is the perfect square of 6 then you can either continue using FOIL success is guaranteed or you can use the quadratic formula for factoring To use the quadratic formula for factoring 1 Find the roots of the equation ax2 bx c 0 b r m x1 x2 2a 2 Factor the trinomial according to the formula ax2 bxc ax xlx x2 You can distribute a and end up with two factors NOTE You can factor ANY trinomial using the Quadratic Formula Exercises Factor the trinomials 1 6m2 l3m 5 2 10x2 2lx9 3 6x27x 3 4 3y2yz lOZ2 5 l2xz7x l2 6 30x2 65x10 7 2t2 3t l 8 x2 2x 63 9 400x2l20x9 10 12x2 l6x 3 Answers l3m l2m 5 2 5x 32x 3 33x l2x3 4 3 y 52y 2z LEI 3 4 4 53x 44x 3 6 56x lx 2 7 2t 8 x 9x7 9 20x 32 perfect square 10 6x l2x 3 LECTURE OUTLINE L1 Rectangular Coordinates Review A1 Section 11 Review Classify various sets of numbers Identify the sets of natural integer rational irrational and real numbers Compute absolute value of a number or expression Find the distance between two numbers on the number line De ne all square roots Evaluate the principal square root Skills 1 Plot points in the rectangular coordinate system Draw a scatter diagram Use a graphing utility to draw a scatter diagram 2 Compute the distance between two points in the coordinate plane Verify right triangles 3 Find the coordinates of the midpoint of a line segment Solve applied problems using the midpoint L2 Graphs of Equations Review A5 Section 12 Review Solve linear equations and equations that lead to linear equations Solve an equation for the indicated variable Solve quadratic equations by factoring and using the square root method Solve quadratic equations by using the quadratic formula Use the discriminant to classify the types of solutions of a quadratic equation Skills 1 Define an equation in two variables and its graph Graph some basic equations by plotting points 2 Find the intercepts of the graph of an equation 3 Define the types of symmetry with respect to the XaXis the yaXis and the origin Test an equation for symmetry L3 Circle Review A3 A5 Section 12 Review Expand or factor polynomials by using the special products Factor out the common factor Factor polynomials by grouping Define prime irreducible polynomials Factor polynomials over the various sets of numbers integer rational or real numbers Factor a second degree trinomial by using FOIL in reverse Use method of substitution for factoring polynomials Solve quadratic equations by completing the square Skills 1 Write equations of a circle in various forms 2 Find the center and radius of a circle from the given equation 3 Graph a circle L4 Lines Section 13 Skills 1 De ne and interpret the slope of a nonvertical line Find the slope of a nonvertical line Graph lines given the point and the slope Consider vertical lines for which the slope is unde ned 2 Analyze the graph of a line using the slope Graph a line 3 Consider various forms of equations of lines such that pointslope slopeintercept and general form 4 Give criteria for parallel and perpendicular lines Find equations of parallel and perpendicular lines Consider the special cases of horizontal and vertical lines 5 Interpret of the slope as a ratio and as a rate of change L5 Relations and Functions Review A6 Section 14 Review State the properties of inequalities Use these properties to solve linear inequalities Solve double inequalities Solve inequalities involving absolute value Skills 1 Give the de nition of a relation Distinguish functions from general relations Discuss the four ways of representing a function 2 Determine whether a relation represents a function 3 Evaluate functions Simplify the difference quotient 4 Introduce a piecewisede ned function De ne the domain of a function 5 Solve applied problems involving constructing and analyzing a function L6 Analyzing Graphs of Functions Section 15 Skills De ne the graph of a function Find the domain and range of a function from its graph Identify the graph of a function by using the vertical line test Find zeros of a function Determine using a graph where the function is increasing decreasing and constant De ne relative maximum and minimum and nd them from a graph Approximate a relative maximumminimum by using a graphing utility Introduce the average rate of change Discuss its geometrical meaning De ne even and odd functions and their properties Identify even odd functions Analyze the graph of a function pW QMrbE Nf L7 Library of Functions Section 16 Skills 1 Introduce linear functions Discuss their properties Identify linear functions by using average rate of change 2 State when a linear function is increasing decreasing or constant 3 Discuss properties and the graph of the reciprocal function square root function and squaring function 4 Analyze and graph piecewisede ned functions Investigate the properties and graphs of absolute value and greatest integer functions 5 Consider applied problems on constructing a piecewisede ned function L8 Graphing Techniques Transformations Review A5 Section 17 Review Solve equations quadratic in form Solve biquadratic equations Solve equations with absolute value Skills 1 Consider Rigid Transformations vertical and horizontal translations and re ections across the xaxis and the yaxis 2 Consider Nonrigid Transformations vertical and horizontal compressions and stretches 3 Obtain new functions from the basic ones using the transformations 4 Consider application problems involving transformations L9 Arithmetic Combinations of Functions Review A2 A5 Section 18 Review De ne the principle nth root of a real number Evaluate radical expressions Consider cancellation rules for radicals and exponents Defrne rational exponents Simplify expressions with radicals andor rational exponents Solve equations with radicalsrational exponents Skills 1 Form sum difference product and quotient of two functions 2 Find the domain of sum difference product and quotient of two functions L10 Composition of Functions One to one Functions Sections 18 19 Skills 1 De ne a composite function Discuss its domain 2 Find the composite function and its domain 3 Evaluate a composite function given by tables 4 Find the components of a composite function 7 decompose the function 5 Consider applications involving composite functions 6 De ne inverse relations 7 Discuss on what conditions an inverse of a function is a function itself Relate your discussion to a onetoone function 8 Use a horizontal line test and analytic de nition to determine whether the function is onetoone 9 Make a function onetoone by restricting its domain L11 Inverse Functions Mathematical Modeling and Variation Sections 19 110 Skills 1 De ne the inverse of a given function Determine whether two given functions are inverses of each other 2 Apply the cancellation rules for the inverse functions State the relation between the domain and range of the function and its inverse Obtain the graph of the inverse function from the graph of a function 3 Find the inverse of a onetoone function defined by an equation Find the inverse of a domainrestricted function 4 Construct linear models Use the linear regression feature to nd the line of best t Discuss the meaning of the correlation coef cient r 5 De ne different types of variation Construct mathematical models using direct inverse and joint variations L12 Quadratic Functions and Models Review A3 Section 21 Review De ne polynomials in one variable Determine whether a given function is a polynomial Skills 1 Introduce a polynomial function 2 De ne a quadratic function Graph quadratic functions by using transformations 3 Discuss properties of a quadratic function 4 Graph a quadratic function by using the vertex axis and intercepts 5 Analyze a quadratic model Find the maximumminimum value of a quadratic function 6 Use a graphing utility to construct a quadratic function of best t from data and analyze its properties L13 Polynomial Functions Section 22 Skills 1 Give the de nition and discuss the features of a polynomial function 2 Analyze the properties and the end behavior of a power function 3 De ne a zero and its multiplicity Find all zeros of a polynomial written in factored form and give the multiplicity of each zero 4 Debate how the multiplicity of a real zero predef1nes the polynomial s behavior near that xintercept 5 De ne the turning points State the relation between the number of turning points of the polynomial and its degree 6 Discuss the end behavior of a polynomial Stress on the sign of the leading coef cient to determine the end behavior 7 Give an equation of a polynomial function which has the given graph 8 Consider applications of a polynomial function L14 Graphing Polynomial Functions Polynomial Inequalities Polynomial Division Sections 22 27 23 Skills 1 Analyze a polynomial function and draw its graph 2 Use a graphing utility to analyze and graph a polynomial function 3 Discuss the technique for solving polynomial inequalities Solve polynomial inequalities 4 Construct a polynomial cubic function of best t 5 Perform long division on the polynomials 6 Perform synthetic division where it is appropriate L15 Complex Numbers and Quadratic Equations Section 24 Skills 1 De ne the complex numbers Equate add subtract multiply and divide complex numbers Simplify powers of 139 2 De ne the principal square root of a negative number Simplify expressions involving roots of negative numbers 3 Use the quadratic formula for the equations with a negative discriminant Solve quadratic equations in the complex number system 4 Solve equations that lead to the quadratic equations L16 Zeros of Polynomial Functions Sections 23 22 25 Skills 1 State the division algorithm for polynomials Use the Remainder and Factor Theorems Use the Conjugate Pairs Theorem 2 State the Fundamental Theorem of Algebra Prove the Linear Factorization and Number of Zeros Theorems Factor polynomials by using its zeros 3 Find all zeros of a polynomial function when one zero is given Find a polynomial with speci ed zeros 4 State the Intermediate Value Theorem and use it to locate a real zero of a polynomial 5 Use the Theorem for Bounds on Zeros Omit Descartes Rule of Signs Rational Zeros Theorem 6 Use a graphing utility to approximate the real zeros of a polynomial function L17 Rational Functions Asymptotes and Holes Section 26 Skills 1 Identify a rational function and nd its domain De ne the real zeros of a rational function 2 De ne vertical asymptotes and holes of a rational function Find the vertical asymptotes and the holes if any 3 De ne horizontal and oblique asymptotes of a rational function Find the horizontal or oblique asymptote of a rational function 4 Solve applied problems involving asymptotes of rational functions L18 Graphing Rational Functions Rational Inequalities Sections 26 27 Skills 1 Graph rational functions Construct a rational function from the given graph 2 Solve applied problems involving rational functions 3 Solve rational inequalities L19 Systems of Linear Equations Substitution and Elimination Systems of Nonlinear Equations Sections 71 72 Skills 1 Identify systems of two equations in two variables De ne the solution set of the system of equations and discuss its geometrical meaning Discuss the possible solution sets of the system of two linear equations in two variables 2 Solve systems of two linear equations in two variables by substitution and elimination 3 Solve applied problems by using systems of two linear equations in two variables 4 Solve systems of nonlinear equations by substitution or elimination Graph the equations of the system and locate its solutions Use a graphing utility to approximate the solutions of a system of two equations in two variables 5 Use a graphing utility to solve applied problems involving systems of two equations in two variables L20 Exponential Functions Section 31 Skills 1 De ne the exponential function Compare exponential and linear models 2 Graph exponential functions Analyze the properties of exponential functions 3 Solve exponential equations using onetoone property of the exponential function 4 De ne number 6 Graph the exponential function with the base 6 5 Use the regression feature of a graphing calculator to nd the exponential function of best t L21 Logarithmic Functions Applications of Logarithms Sections 32 35 Skills 1 De ne the logarithmic function State the cancellation rules for logarithms and exponents Convert between logarithmic and exponential expressions Solve logarithmic and exponential equations by converting from one form to another 2 Graph the logarithmic function Analyze the properties of the logarithmic function knowing the properties of the exponential function 3 Give a de nition of log x Evaluate logarithms with and without a calculator 4 De ne the natural logarithmic function Graph natural logarithmic functions 5 Consider applications of logarithms L22 Properties of Logarithms Logarithmic and Exponential Equations Sections 33 34 Skills 1 State the properties of logarithms Simplify expressions by using the properties of logarithms Use the changeofbase formula 2 Solve logarithmic equations 3 Solve exponential equations 4 Use a graphing utility to approximate the solutions of exponential and logarithmic equations L23 Applications of Exponential and Logarithmic Functions Logistic Models Building Models from Data Sections 31 35 Skills 1 State the compound interest formula Consider continuous compounding 2 Determine the present and the future value of a lump sum of money 3 Determine the rate of interest and time required to obtain a certain amount of money 4 Find equations describing the quantities that obey the law of uninhibited growth and decay Find the halflife of a radioactive substance 5 Use logistic models 6 Use a graphing utility to t exponential logarithmic and logistic functions to data L24 Mathematical Induction the Binomial Theorem Sections 94 95 Skills 1 Prove statements using mathematical induction 2 Evaluate 3 State the Binomial Theorem Expand binomials Evaluate a speci ed coef cient in a binomial expansion 4 Describe the Pascal s triangle Use the Pascal s triangle to write the binomial expansions for the small natural numbers n FINAL EXAM L1 L24
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