Instructor: Ann Ostberg
Semester: Spring
This course is designed to explore the concepts of college algebra. It will include a study of linear, quadratic, rational, polynomial, and radical equations; relations and functions; rectangular coordinate system and graphs; systems of equations and inequalities; exponential and logarithmic functions; and matrices.
·
College Algebra (with Interactive
Video Skillbuilder CD-ROM and CengageNOW, iLrn™ Tutorial Student Version,
Personal Tutor with SMARTHINKING Printed Access Card), 9th Edition;
Gustafson/Frisk; ISBN-10: 0-495-01266-1 or ISBN-13: 978-0-495-01266-5
Thomson © 2007
Unit |
Topic |
Learning Objectives |
1 |
Welcome |
· This section reviews the different sets of numbers such as natural numbers, whole numbers, integers, and rational numbers and their properties. It continues with graphing on a number line and associated topics such as intervals, inequality symbols, absolute value, and distance. · This section reviews exponents and the rules of exponents along with order of operations, evaluating expressions, and scientific notation. |
2 |
rational exponents, radicals, and polynomials |
· This section reviews rational exponents. Rational exponent is another name for fractional exponents. Radical expressions such as square roots, cube roots, etc. are presented along with the methods to simplify and combine (addition and multiplication). Remember rationalizing the denominator? We will also rationalize the numerator in this section. · This section covers polynomials. We will define, add, subtract, multiply, and divide. Remember the term FOIL? A new term might be conjugate binomials. |
3 |
factoring of the polynomials and algebraic fractions |
· This section lays the foundation of a key process in algebra: factoring polynomials. Many factoring techniques will be presented. Practice as much as possible to become proficient! · This section reviews the basic concepts of fractions; however, it extends it to algebraic fractions. The techniques to simplifying, multiplying, dividing, adding, and subtracting algebraic fractions are based on simple fractions that you learned in grade school. To make things more interesting, we will also work with complex fractions. |
4 |
equations and their applications |
· In this section, you’ll learn about properties of equality, linear equations, rational equations, and formulas. All of these concepts are vital to your ability to work with equations. · This section extends the concepts of linear equations to the practical application of linear equations. You may be familiar to these as ‘story problems.’ Pay close attention to the Strategy for Modeling with Equations in your text. |
5 |
quadratic equations and their applications and complex numbers |
· This section introduces quadratic equations. These are second-degree equations (they have an exponent of 2 on the variable). The key topics are solving the quadratic equation, i.e., finding the values that make the equation a true statement. In order to solve, the methods of zero-product, completing the square, square root property, and the quadratic formula will be used. · This section looks at some applications of quadratic equations. We will place emphasis on the geometric problems, uniform motion problems, and flying object problems. · This section investigates complex numbers. Complex numbers developed as solutions to certain quadratic equations. There are two types of complex numbers: real and imaginary. We will be looking at the definition and simplification of imaginary numbers and complex numbers. The techniques of FOIL will prove useful as you do arithmetic with complex numbers. |
6 |
polynomial and radical equations and inequalities |
· This section teaches solving polynomial equations by factoring and the methods used in solving radical equations. So your factoring skills will be put to the test in this section. Radical equations aren’t trying to protest anything but they are equations that involve square roots and cube roots. · This section looks at the properties of inequalities. You will utilize those properties to solve linear inequalities and compound inequalities. Of particular emphasis will be solving quadratic and rational inequalities. Of these types, you must make a separate chart to actually find the solution. |
7 |
absolute value |
· This section on absolute value concludes the chapter. Emphasis will be placed on the definition of absolute value and solving equations and inequalities involving absolute value. We will look at equations with two absolute values – these are actually easier than they sound. |
8 |
rectangular coordinate system |
· In this section, you will be able to graph linear equations on the Rectangular Coordinate Plane. On this plane, one may also find the distance and/or the midpoint between two points. Some basic applications will be presented. Be sure to understand these formulas and the terminology associated with this topic. · This section focuses on the slope of lines. This will include horizontal, vertical, perpendicular, and parallel lines. An emphasis will be placed on the nonvertical lines (simply all lines that are not vertical). You will want to memorize the formula for slope. · This section focuses on the methods of taking graphical information (points, slope, etc) and translating that into an algebraic equation. This was actually a very revolutionary development in the history of mathematics! Key formulas are the point-slope form and the slope-intercept form. |
9 |
graphs of equations and ends with proportions and variations |
· This section covers the graphing of other types of equations. Emphasis will be placed on the symmetries of graphs, miscellaneous graphs such as absolute value, square root, quadratic (parabolas), and circles. The formula for a circle will be presented. · This section presents the topics of proportion and variation. You have undoubtedly worked proportion problems in earlier math classes. You will use the process of ‘cross-multiplication.’ Variation problems might be a bit new, however, they are practical problems. Direct variation implies that as one element (x) increases so does the other (y). Think of as you eat more calories, you will gain more weight. Indirect or inverse variation implies that as one element increases (x) the other (y) decreases. |
10 |
functions and function notation and quadratic functions |
· This section begins the discussion on functions. The variables, x and y, will be given new ideas: independent variable and dependent variable for x and y, respectively. The x values will also be referred to as the domain. The y values will be considered the range of the function. Understand the notation that is used to denote a function. We will conclude with drawing the graphs of functions: actually you have already done this. Using the ‘vertical line test’ will allow you to tell whether a graph is the graph of a function. · This section looks specifically at the quadratic function. You are already familiar with this function as its graph is a parabola. The quadratic equation is very important as it has many practical applications. Besides knowing how to graph the parabola, finding the vertex is very useful. The vertex will tell the maximum or minimum value of a situation. |
11 |
polynomials and other functions. Translating graphs and rational functions |
· This section focuses on graphing polynomial functions. Did you know that there were ‘even’ and ‘odd’ functions? We will also look at increasing and decreasing function. Think of going up a hill versus going down a hill. Finally, we will graph piecewise-defined functions. These functions often frustrate students until they realize that it is just like taking a piece from two or more pies (graphs), say apple and cherry, and placing them on your plate (coordinate plane), side by side. · This section will demonstrate how one can move graphs of equations to the right, left, up, down, and flip! These are called translations. The ‘flip’ is called a reflection about the x- or y-axes. For all of these translations, they will start with the basic graph of the equation. · This section will provide a brief introduction to rational functions. The definitions of asymptotes, rational function, and their related domain will be discussed. We will not focus on the graphing of rational equations. |
12 |
functions and inverse functions |
· This section focuses on special functions called inverse functions. Only one–to-one functions can have inverse functions. Visually, this can be determined from the graph of the function using The Horizontal Line Test. You will also learn to write the ‘inverse’ of an equation. |
13 |
exponential functions and their graphs, applications of exponential functions, and logarithmic functions |
· This section will look at a special type of function, called an exponential function. You will always be able to recognize an exponential function as the variable (x) is the exponent. For example, , where b is a constant. The graphs of all exponential functions are similar in shape and go through the point (0, 1). A practical application of exponential functions is finding the value of compound interest. If the interest is compounded continuously, a special number, e, is used. The value of e is 2.7182818.... · This section discusses some of the applications of exponential functions. We will focus only on radioactive decay and Malthusian population growth. · This section covers logarithmic functions and their graphs. A logarithmic function is simply the inverse of an exponential function. The inverse is more commonly written as. If the base (b) is 10, it is called a common logarithm. If the base (b) is e, it is called a natural logarithm. The shape of the graph is similar to the exponential function, except it passes through the point (1, 0) instead of (0, 1). These graphs can also be written so that they have horizontal and vertical movement. |
14 |
logarithmic functions, properties of logarithms, and exponential and logarithms equations |
· This section focuses on the applications of logarithmic functions. We will only look at those applications from electrical engineering, geology, and population growth. · This section investigates the properties of logarithms. The key thing to remember is that a logarithm IS an exponent! Remember the properties associated with exponents and you will see how they become applicable to the properties of logarithms. The Change-of-Base Formula allows one to calculate the value of any logarithm. · This section shows how one may solve exponential and logarithmic equations. The steps are straightforward. Be sure to be able to write equations in exponential form to logarithmic form and vice versa. Carbon-14 dating is an important application that involves logarithmic equations. |
15 |
linear equations, determinants, and graphs of linear inequalities |
· This section covers the various methods that one can use to solve systems of linear equations in two variables: the graphing method, the substitution method, and the addition method. We will also look at the characteristics of a system with either infinitely many solutions or no solutions (inconsistent). Finally, the techniques will be utilized to solve a system of linear equations in three variables. Systems of linear equations are crucial in solving linear programming problems. · This section investigates another method of solving a system of linear equations. It involves the use of matrices. Through the use of determinants (which are found from a matrix), one can find the solution to systems. Techniques involving determinants can also be used to find the equation of a line and the area of a triangle. · The final section of this semester will discuss graphing linear inequalities. You will utilize the techniques of graphing equations and then determining which sections of the graph represent the solution area. Graphing a system of linear inequalities is also used in linear programming. |