College Algebra
LECTURE
1.1 FUNCTIONS , GRAPHS
- Graphs , tables and equations that represent functions .
- Evaluating functions .
- Stock market ; men in workforce .
* Data - Scatter plot – Inputs , Outputs - Function ( Graphically , Numerically , Analytically defined ) – Domain and Range - Functional Notation - The conditions of domains .
1.2 GRAPHS - MATHEMATICAL MODELS
- Graphing and evaluating mathematical models .
- Cost-Benefit , Bankruptcies .
* Point-Ploting methods – Graphing - Mathematical Models - Aligning Data - Evaluating functions with technology – Scatter Plots .
1.3 LINEAR FUNCTIONS
- Intercepts and slopes .
- Constant rates of change .
- Loan Balances , Temperature measurements .
Linear functions – x and y-intercepts – Slope of the line - Linear Models - Slope-Intercept form – Rate of change – Special Functions .
1.4 EQUATIONS OF LINES
- Equations of lines .
- Constant and average rates of change .
Equations of Lines – Slope-intercept form - General form – Point -Slope form - Linear Models - Constant Rates of change – Average Rates of change .
1.5 SOLUTIONS OF LINEAR EQUATIONS
- Solutions , zeros and x-intercept .
- Solving linear equations .
- Functional forms of equations in two variables .
- Credit card debt , Simple interest .
Solution of linear equations algebraically – graphically – Solution of equations for a specified linear variable .
1.6 FITTING LINES - MODELING LINEAR FUNCTIONS
- Linear Regression .
- Modeling and applying with linear equations .
Fitting lines to data points – Linear Regression ( Least squares method ) – Applying Models – Constant first Difference .
1.7 MODELS IN BUSINESS AND ECONOMICS
- Modeling Revenue , Cost and Profit .
- Break Even – Marginal Revenue , Cost and Profit , Supply and Demand .
- Market Equilibrium .
- Break Even in Production and Sales , Marginal Revenue and Cost , Market Equilibrium .
Total Revenue – Total Cost – Profit – Marginal Profit – Marginal Revenue – Marginal Cost – Break Even
1.8 SOLUTIONS OF LINEAR INEQUALITIES
- Solving linear inequalities .
- Solving double (system) inequalities .
- Profit , Body temperature .
Linear inequalities – Algebraic solution of an inequality – Graphical solution of an inequality - Double Inequalities .
LECTURE
2.1 QUADRATIC FUNCTIONS , PARABOLAS
Topics :
- Quadratic function.
- Vertex of Parabola .
- Zero of Quadratic function .
Applications :
- Maximum revenue from sales .
Key Concepts :
*Quadratic function – Parabola – Zero of Quadratic function.
LECTURE
2.2 QUADRATIC EQUATIONS
Topics :
- Solving Quadratic equations by algebraic method .
- Solving Quadratic equations by graphic method .
Applications :
- Profit , Break-Even .
Key Concepts :
* Solving Quadratic equations by graphic – algebraic method .
LECTURE
2.3 POWER FUNCTIONS AND TRANSFORMATIONS
Topics :
- Power functions and root functions .
- Increasing and decreasing functions .
- Symmetry .
- Transformations of graphs .
Applications :
- Magnetude of stimulus and response .
- Mutual funds .
Key Concepts :
* Power functions –Increasing and decreasing – Root functions – Transformations ( Shift , Compression , Stretch , Reflection )
- Radical equations .
LECTURE
2.4 QUADRATIC AND POWER MODELS
Topics :
- Quadratic functions.
- Comparing linear and quadratic models .
- Modeling with power functions .
- Comparing quadratic and power models.
Applications :
- Auto noise .
- Voting .
Key Concepts :
* Modeling quadratic functions - power functions – Comparison of models – First and second differences - Comparison of quadratic and power models
LECTURE
2.5 RECIPROCAL , ABSOLUTE VALUE – PIECEWISE -DEFINED FUNCTIONS
Topics :
- Reciprocal functions .
- Absolute value and Piecewise-defined functions .
Applications :
- Cost - Benefit .
- Average cost .
- Residental power cost .
Key Concepts :
* Reciprocal functions – Average cost function – Piecewise-defined functions – Absolute value functions – Solving absolute value equations.
LECTURE
2.6 INVERSE FUNCTIONS
Topics :
- Inverse functions.
- Finding and graphing the inverse of a functions .
Applications :
- Loan balances .
- Money conversion .
Key Concepts :
* Inverse functions – Horizontal line test – Finding the inverse functions – Graphs of inverse functions.
LECTURE
2.7 QUADRATIC AND POWER INEQUALITIES
Topics :
- Solving quadratic inequalities analytically and graphically .
- Solving power inequalities .
Applications :
- Internet users .
- Investment.
Key Concepts :
* Quadratic inequalities - Solving analytically –graphically – Power inequalities .
3.1 EXPONENTIAL FUNCTIONS
- Exponential functions .
- Transformation of graphs .
- Spreadsheet and the number e .
* Exponential growth function – Horizontal asymptote – Transformation of graphs of exponential functions – The number e .
3.2 LOGARITHMIC FUNCTIONS
- Logarithmic functions .
- Common and natural logarithms .
- Doubling time for investments .
* Logarithmic functions – Base – logarithm – Exponent - Exponential and logarithmic forms - Common and natural logarithms - Change of base .
3.3 EXPONENTIAL EQUATIONS – PROPERTIES OF LOGARITHMS
- Logarithmic properties .
- Solving exponential and logarithmic equations .
- Exponential inequalities .
- Measuring earthquake intensity .
* Solving exponential equations–Properties of Logarithms .
3.4 EXPONENTIAL AND LOGARITHMIC MODELS
- Modeling with exponential equations .
- Exponential and Logarithmic models .
- Comparison of models .
* Modeling with exponential equations – Comparing quadratic and exponential models - Logarithmic modeling .
3.5 EXPONENTIAL FUNCTIONS AND INVESTING
- Continuous compounding .
- Investment models .
-Future value of an account .
- Continuous compounding .
- Present value of an investment .
- Mutual fund growth .
* Future value of investment – Interest compounded annually – Interest compounded k-times per year - Interest compounded continuous - Present value of an investment – Investment models .
- Future value and Present value of an annuity .
- Future value of an ordinary annuity .
- Present value of an ordinary annuity .
* Annuity – Ordinary Annuity – Present value of an investment - Present value of an ordinary annuity - Loan Repayment – Amortization .
3.7 LOGISTIC AND GOMPERTZ FUNCTIONS
- Logistic growth and decay functions .
- Gompertz functions .
* Logistic functions – Limiting value of logistic functions – Modeling logistic functions – Gompertz functions .
LECTURE
4.1 POLYNOMIAL FUNCTIONS
Topics :
- Polynomial functions .
- Cubic functions .
- Quartic functions .
Applications :
- United Nations debt .
- Foreign born population .
Key Concepts :
* Polynomial functions –Cubic functions - Quartic functions – Local extrema ( local minimum , local maximum , absolute minimum, absolute maximum – Increasing and decreasing functions .
LECTURE
4.2 CUBIC AND QUARTIC FUNCTIONS
Topics :
- Modeling with cubic and quartic functions .
- Model comparison .
Applications :
- Sales of tobacco .
- US foreign born population .
Key Concepts :
*Cubic models – Quartic models .
LECTURE
4.3 POLYNOMIAL EQUATIONS
Topics :
- Solving polynomial equations .
- Estimating solutions with technology .
Applications :
- Cost .
- FV of an investment .
- Stock prices .
Key Concepts :
* Solving polynomial equations – Estimating solutions with technology .
LECTURE
4.4 SOLUTIONS OF POLYNOMIAL EQUATIONS
Topics :
- Solving cubic and quartic equations .
- Combining graphical and analytical methods .
Applications :
- Break-even revenue .
Key Concepts :
* Polynomial division – Using division to solve cubic equations – Combining graphical and analytical methods – Solving quartic equations .
LECTURE
4.5 COMPLEX SOLUTIONS - FUNDAMENTAL THEOREM OF ALGEBRA
Topics :
- Fundamental Theorem of Algebra .
- Imaginary and complex numbers .
- Operations with complex numbers .
- Quadratic and Polynomial equations with the complex solutions .
Applications :
- Fractal .
Key Concepts :
* Imaginary and complex numbers – Operation with complex numbers – Quadratic equations with the complex solutions – Polynomial equations with complex solutions – Fundamental Theorem of Algebra .
LECTURE
4.6 RATIONAL FUNCTIONS AND RATIONAL EQUATIONS
Topics :
- Rational functions .
- Solutions of rational equations .
Applications :
- Average cost of production .
- Cost - Benefit .
- Advertising and Sales .
Key Concepts :
* Graph of rational functions – Vertical asymptotes – Horizontal asymptotes – Slant asymptotes – Solving the rational equations ( analytically and graphically ) .
LECTURE
4.7 POLYNOMIAL AND RATIONAL INEQUALITIES .
Topics :
- Polynomial inequalities .
- Rational inequalities .
Applications :
- Average cost of production .
Key Concepts :
* Polynomial inequalities ( graphical solutions , analytical solutions )– Rational inequalities ( graphical solutions , analytical solutions ) .
LECTURE
5.1 SYSTEMS OF LINEAR EQUATIONS IN 2 VARIABLES
Topics :
- Graphical solution of systems .
- Modeling systems of linear equations .
- Inconsistent and dependent systems .
Applications :
- Break-even .
- Market equilibrium .
- Investing .
Key Concepts :
* Solution of systems of equations ( by graphing , substitution , elimination )– Type of systems ( unique solutions , dependent systems , inconsistent systems ) – Market equilibrium ( demand , supply ) – Modeling systems of linear equations .
LECTURE
5.2 SYSTEMS OF LINEAR EQUATIONS IN 3 VARIABLES - MATRIX SOLUTION
Topics :
- Left-to-right elimination method .
- Matrix representation of systems of linear equations .
- Solving systems with matrices.
- Gauss-Jordan elimination method .
Applications :
- Product transportation .
- Manufacturing .
- Investment .
Key Concepts :
* Left-to-right elimination method – Matrix representation of systems of linear equations ( matrix , dimensions , square matrix , indentity matrix , augmented matrix , coefficient matrix ) – Gauss-Jordan elimination method – Solution with technology .
LECTURE
5.3 SYSTEMS OF LINEAR EQUATIONS WITH NON-UNIQUE SOLUTIONS
Topics :
- Non-unique solutions .
- Dependent systems of linear equations .
- Inconsistent systems .
Applications :
- Purchashing .
- Investment .
- Transportation .
Key Concepts :
* Non-unique solutions – Dependent systems – Inconsistent systems .
LECTURE
5.4 MATRICES – BASIC OPERATIONS
Topics :
- Addition and substraction of matrices .
- Multiplication of a matrix by a number .
- Matrix multiplication .
- Multiplication with technology .
Applications :
- Pricing .
- Trade Balance .
- Advertising .
- Manufacturing .
Key Concepts :
* Matrix operations ( Addition , Substraction , Multiplication by a constant – Matrix multiplication ) .
LECTURE
5.5 INVERSE MATRICES – MATRIX EQUATIONS
Topics :
- Inverse matrices .
- Inverse and technology .
- Solving matrix equations .
- Matrix equations with technology .
Applications :
- Manufacturing .
- Investment .
Key Concepts :
* Inverse matrices - Inverse and technology - Solving matrix equations .
LECTURE
6.1 SYSTEMS OF LINEAR INEQUALITIES
Topics :
- Linear inequalities in 2 variables .
- Systems of linear inequalities in 2 variables .
Applications :
- Auto purchases .
- Car rental .
- Advertising .
Key Concepts :
* Linear inequalities in 2 variables ( Solution region , test point ) – Systems of linear inequalities in 2 variables ( border , corner ) .
LECTURE
6.2 LINEAR PROGRAMMING - GRAPHICAL METHODS
Topics :
- Linear programming .
- Solutions with technology .
Applications :
- Maximizing Car rental profit .
- Cost Minimization .
- Profit .
Key Concepts :
* Linear programming – Constraints – Objective function – Feasible solution – Feasible region – Optimum value ( Maximum , minimum ) .
LECTURE
6.3 SEQUENCES - DISCRETE FUNCTIONS
Topics :
- Sequences .
- Arithmetic Sequences .
- Geometric Sequences .
Applications :
- Investment .
- Depreciation .
- Compound Interest .
Key Concepts :
* Sequences – Arithmetic Sequences – Geometric Sequences .
LECTURE
6.4 SERIES
Topics :
- Finite and Infinite Series .
- Arithmetic and Geometric Series .
- Infinite Geometric Series
Applications :
- Profit .
- Depreciation .
- Annuities .
Key Concepts :
* Finite Series – Infinite Series - Sigma Notation - Arithmetic Series – Geometric Series - Infinite Geometric Series .
LECTURE
6.5 PREPARING FOR CALCULUS
Topics :
- Simplifying expressions and fractions .
- Evaluating Functions .
- Rewriting Radical .
- Decomposition of Functions .
- Rewriting Expressions using negative and positive exponents .
- Using logarithmic properties .
- Reducing Fractions .
Applications :
- The equation of tangent line .
Key Concepts :
* Chapter 1 Skills – Chapter 2 Skills – Chapter 3 Skills – Chapter 4 Skills – Chapter 5 Skills .