Course Title and Description:
MTH 158 - College Algebra
Covers the structure of complex number systems, polynomials, rational expressions, graphing, systems of equations and inequalities and functions, quadratic and rational equations and inequalities.
Lecture 3 hours per week.
3 Credits
Prerequisites: Competency in Math Essentials MTE 1-9 as demonstrated through the placement and diagnostic tests, or by satisfactorily completing the required MTE units or equivalent. (Credit will not be awarded for both MTH 163 and MTH 166.)
Required Prerequisite Knowledge: To succeed in this course a student must have succeeded in secondary-level mathematics courses through Algebra II. A student should understand that this course is somewhat more challenging than most secondary-level algebra coruses, and will require more work than these courses.
Secondary-level Algebra I and Algebra II courses are considered equivalent to MTE 1 - 9.
Topics in the MTE 1-9 courses include the following, and students are expected to begin the course with exposure to all these topics. The course also begins with a chapter-long review of most of these topics.
MTE 1 - Operations with Positive Fractions
MTE 2 - Operations with Positive Decimals and Percents
MTE 3 - Algebra Basics
MTE 4 - First Degree Equations and Inequalities in One Variable
MTE 5 - Linear Equations, Inequalities and Systems of Linear Equations in Two Variables
MTE 6 - Exponents, Factoring and Polynomial Equations
MTE 7 - Rational Expressions and Equations
MTE 8 - Rational Exponents and Radicals
MTE 9 - Functions, Quadratic Equations and Parabolas
This course is offered via the Internet using structured problem sets and the required textbook. The student may also choose to use the recommended supplement Math-XL, though this is optional and is not part of the course requirement and/except for the preparation provided by Math-XL, has no bearing on grades.
The student will receive instructional information and assignments via the course webpage and will respond to assignments by submitting work through web forms.
The student must have standard access to the Internet.
The instructor is available via web forms (to which students will be introduced at the very beginning of the course), and will normally respond by the end of the day following your submission (and often on the same day) with answers to properly posed questions, feedback on your efforts, and other information. Exceptions may occur in the event of Internet problems or other technical events, and at times of excessive demand.
Broad goals and Purpose of the CourseThe broad goals and purpose of the course include the following:
To gain a conceptual understanding of and the ability to use mathematical functions in a real-world context, utilizing algebraic techniques (including but not limited to computer algebra software) and visualization (using but not limited to computer or calculator graphing technology), while working and communicating in a cooperative and collaborative effort to document the learning process and its end results.
Understanding of the nature of the mathematical modeling process, its uses and its limitations.
Proficiency in mathematical modeling using various functions, including but not limited to linear, quadratic, power and polynomial functions.
Specific objectivesEach assigned task and problem constitutes a specific objective, which is to complete that problem or task and understand as fully as possible its relationship to the stated goals of the assignment and to other concepts, problems and situations encountered in the course.
More specifically, the following objectives are to be achieved.
Know and apply standard set terminology.
Classify subsets of real numbers.
Evaluate numerical expressions using correct order of operations.
Apply the reflexive, symmetric, transitive properties and the principle of substitution.
Apply the commutatve, associative, distributive and identity properties.
Find additive inverses and reciprocals.
Apply the rules of signs.
Apply the zero-product property.
Graph a given inequality.
Find the distance between two points on the real number line.
Evaluate a given algebraic expression for a given value of the variable.
Determine the domain of the variable for a given algebraic expression.
Apply the laws of exponents.
Evaluate square roots of various expressions.
Express decimal numbers in scientific notation and numbers given in scientific notation as decimals.
Apply the Pythagorean Theorem and its converse.
Apply basic formulas for areas and volumes of geometric regions.
Know and apply the conditions for congruence and similarity of triangles.
Distinguish between monomials and polynomials.
Add, subtract and multiply polynomials.
Know and apply formulas for special products.
Divide polynomials using long division.
Factor the difference of two squares and the sum and difference of two cubes.
Factor perfect squares.
Factor second degree polynomials.
Factor by grouping.
Reduce a rational expression to lowest terms.
Multiply and divide rational expressions.
Add and subtract rational expressions.
Simplify complex rational expressions.
Know and apply the meaning of the nth root of a numerical or algebraic expression.
Express the nth root of a rational expression using rational exponents.
Express radical expressions using rational exponents.
Simplify radical expressions, expressing the results in standard form with rational exponents.
Solve linear equations with numerical or symbolic coefficients.
Solve applications problems using linear equations and their solutions.
Solve quadratic equations by factoring.
Solve quadratic equations by completing the square.
Solve quadratic equations using the quadratic formula.
Solve applications problems using quadratic equations and their solutions.
Solve radical equations.
Solve equations which are quadratic in form.
Solve equations by factoring.
Express an interval of the real number line given in interval notation, as a graph or as an inequality in each of the other two ways.
Manipulate inequalities using the addition and multiplication properties of the same.
Solve combined inequalities.
Solve equations and inequalities involving absolute value.
Translate verbal descriptions into mathematical expressions.
Solve interest problems.
Solve mixture problems.
Solve mixture problems.
Solve uniform motion problems.
Solve constant rate work problems.
Apply the distance formula to two points in the plane.
Apply the midpoint formula to two points in the plane.
Demonstrate the relationship between the distance formula and the Pythagorean Theorem.
Apply the distance formula to mathematical or real-world applications.
Graph a given equation by selecting and plotting appropriate points.
Given the graph of an equation find the intercepts.
Given an equation, algebraically determine the intercepts of its graph.
Test an equation for symmetry with respect to the x-axis, y-axis and origin.
Graph the key equations y = x^3, x = y^2 and y = 1 / x.
Calculate and interpret the slope of a line.
Graph lines given a point and the slope.
Find the equation of a vertical line.
Use the point-slope form of a line
Identify horizontal lines.
Find the equation of a line given two points.
Write the equation of a line in slope-intercept form.
Identify the slope and y intercept of a line from its equation.
Graph lines written in general form using intercepts.
Find equations of parallel lines.
Find equations of perpendicular lines.
Given its center and radius, write the equation of a circle in standard form.
Given the equation of a circle in standard form, find its center and radius and graph it.
Graph a circle given its equation in general form.
Draw and interpret the scatter diagram of a given data set.
Distinguish whether the behavior of a scattergram indicates a linear and nonlinear relation.
For a linear relation, sketch a good estimate of the best-fit line and determine its equation.
Construct models using direct, inverse and joint variations.
Apply models of direct, inverse and joint variation to quantify the behavior of the associated system.
Determine whether a given relation represents a function.
Find the value of a function given the value of the independent variable.
Find the domain of a given function.
Determine whether a given graph depicts a function.
Given the graph of a function and value(s) of the independent variable, find the corresponding value(s) of the dependent variable and express the result(s) in function notation.
Given the graph of a function and value(s) of the dependent variable, find the corresponding value(s) of the independent variable and express the result(s) in function notation.
Given the graph of a function determine if the function is an even function, an odd function or neither.
Given the equation of a function determine if the function is an even function, an odd function or neither.
Given the graph of a function determine the interval(s) on which the function is increasing, decreasing or constant.
Given the graph of a function locate its local maxima and minima.
Given the graph of equation of a function determine its average rate of change on a given interval.
Without the use of technology graph the constant function, identity function, square function, cube function, square root function, cube root function, reciprocal function, absolute value function and greatest integer function.
Graph a given piecewise-defined function.
Use the above to solve problems involving mathematical or real-world applications.
Based on the graph of a given function, including but not limited to the constant, identity, square, cube, square root, cube root, reciprocal and greatest integer functions, construct the graph of the given function by applying shifting, stretching/compressing and reflection transformations as appropriate.
Build functions to model various mathematical and real-world problems and situation, analyze the function and interpret the results.
Given a linear function determine its average rate of change.
Given the average rate of change of a linear function and its value corresponding to a given value of the independent variable, find the equation of the function.
Use linear functions to solve problems involving mathematical or real-world applications.
Graph a given quadratic function based on the graph of the squaring function, using appropriate transformations.
Identify the vertex and axis of symmetry of a quadratic function given algebraically or graphically.
Graph a quadratic function given algebraically using its vertex, axis of symmetry and intercepts.
Find the maximum or minimum value of a given quadratic function.
Use quadratic functions to solve problems involving mathematical or real-world applications.
Given a set of nonlinear data construct a scatter plot and sketch the trendline.
Using a trendline with a smoothly varying slope, extend the trendline to its vertex; based on the vertex and a representative point on the trendline determine the approximate quadratic function that models the trendline.
Given a function recognize whether it is or is not a polynomial, and if it is determine its degree.
Apply shifting and stretching transformations and reflections to graph polynomial functions.
Identify the real zeros of a polynomial and their multiplicities.
Analyze the behavior of a polynomial near a given zero, and relate the behavior to the multiplicity of the zero.
Given a graph determine whether it is the plausible graph of a polynomial, and if it is determine its minimum degree.
Relate the degree of a polynomial to its behavior for large | x |.
Find the domain and range of a given rational function.
Find the vertical, horizontal and oblique asymptotes of a given rational function.
Graph a given rational function.
Given a rational function determine its asymptotes and its behavior near its asymptites, its zeros, the intervals on which it is positive or negative, and use the information to sketch its graph.
Use rational functions to solve mathematical and real-world applications.
Determine whether a given function is one-to-one.
Construct the graph of the inverse of a given one-to-one function from the graph of the function.
Construct a table of values for the inverse of a function given a table of its values.
Algebraically find the inverse of a function which is defined algebraically.
Construct the graph of a given exponential function using only two points and the basic properties of an exponential function.
Apply shifting, stretching and reflection transformations to construct graphs of given functions which are based on exponential functions
Define the number e as a limit and show how values of the expression approach the limit.
Solve exponential equations which involve integer powers of the base.
Change exponential expressions to logarithmic expressions and logarithmic expressions to exponential expressions.
Evaluate logarithmic expressions.
Determine the domain of a logarithmic function.
Graph logarithmic functions.
Know and apply the laws of logarithms.
Use logarithms to solve exponential equations.
Solve logarithmic equations.
Solve systems of two or three linear equations in three variables by substitution or by elimination
Identify inconsistent and dependent systems of of two or three linear equations in three variables.
Solve problems involving mathematical or real-world systems of linear equations by writing and solving the appropriate system.
Solve linear systems by writing the appropriate augmented matrix and performing the appropriate set of row operations.
Translate a given augmented matrix into a system of linear equations.
Evaluate 2 by 2 determinants.
Evaluate 3 by 3 determinants by expanding in minors and cofactors on a given row or column.
Expand higher-order determinants.
Use Cramer's Rule to solve a given system of linear equations.
Find scalar multiples of a matrix.
Find a given linear combination of two or more matrices.
Determine whether two matrices are compatible for multiplication in each of the two possible orders.
Know and apply the algebraic properties of matrices (e.g., associative, distributive, identity and inverse properties involving multiplication and addition).
Identify a given matrix as singular and nonsingular by the specified method: by calculating its determinant, or by row reduction, or by representing it as a system of equations and determining the consistency or dependency the system.
Invert a nonsingular square matrix A using row operations on the matrix [ A | I ].
Solve a given system of equations by expressing it as a matrix equation, finding and applying the appropriate inverse matrix.
Regular communication is required of the student. This includes turning in assignments in a timely fashion and responding in a timely manner to feedback on these assignments. Any deviation of more than three days from the chosen schedule of the course must be approved in advance by the instructor. Exceptions will of course be made in the event of documented illness or other unexpected emergencies, but the instructor should be informed of such situations within a reasonable time of occurrence.
After registering for the course you will receive an email, sent to your VCCS email account, with instructions for Orientation and Startup. This process will constitute appropriately the first week's assignments for your course (about the first half of the week during the shorter summer term), and will show you the basic navigation of the website including how to communicate, submit work, locate assignments and due dates, and more.
All assignments and all necessary materials will be available on the homepage. Class notes are included online will be distributed, with additional working video links, in DVD format.
The text is College Algebra by Sullivan, Current Edition. Units covered include Chapters R, 1, 2, 3 and 4 in their entirety, Sections 1-5 on Chapter 5 (Polynomial and Rational Functions), and Sections 1-4 of Chpater 8.
Chapter titles include
Chapter titles include
Students will complete and submit the assignments specified on the homepage.
The instructor will respond in a timely fashion to any work submitted, making suggestions where improvement is needed and posing questions designed to enhance the student's learning experience. The student will be required to respond to all critiques, except those designated otherwise.
Questions posed by students and the instructor's responses will be posted to a site, specified in at the beginning of the course, for the student's review.
Students may on occasion be asked to critique work done by other students. Full student anonymity will be preserved, with no reference to the identity of any party in this exchange.
The instructor is available via web forms (to which you will be introduced at the very beginning of the course), and will normally respond by the end of the day following your submission (and more typically on the same day) with answers to properly posed questions, feedback on your efforts, and other information. Exceptions may occur in the event of Internet problems or other technical events.
Limited use of email: Prior to registration and receipt of initial instructions students my use Email to communicate with the instructor. However email is much less reliable than web forms, and after registration and receipt of initial instructions anything sent through email should first be sent using the appropriate form.
Grading policyFive tests will be administered. The test listed as "Chapter 4 Test" includes the assigned material for both Chapter 4 and Chapter 5. The Chapter 6 test listed on the Testing site is omitted.
A student's portfolio, consisting of instructor responses to assigned work and/or daily quizzes, will at the end of the term be assigned a grade. A student who completes all assigned work in the prescribed manner can expect to make an A on this aspect of the course. The average of grades assigned on this work will count as 1/4 of a test grade. If this average is higher than the average on other tests, it will be counted as 1/2 of a test grade.
Raw test scores will be normalized to the following scale, according to the difficulty of the test, as specified in advance of each test by the instructor:
A: 90 - 100
B: 80 - 90
C: 70 - 80
D: 60 - 70
F: Less than 60.
The final grade will be a weighted average according to the above guidelines. A summary of the weighting is as follows:
Each test: Weight is 1.0
Assignment/Quiz Grade Average: Weight .25 or .5, to the advantage of the student.
Criteria for Grading of Tests:
Tests will consist of problems designed to measure the level of your achievement of the course goals.
Each problem is graded on a 10-point scale, with the following guidelines:
In the event of a college-wide emergency
In the event of a College-wide emergency, course requirements, classes, deadlines, and grading schemes are subject to changes that may include alternative delivery methods, alternative methods of interaction with the instructor, class materials, and/or classmates, a revised attendance policy, and a revised semester calendar and/or grading scheme.
In the case of a College-wide emergency, please refer to the following about changes in this course:
· Instructor’s email dsmith@vhcc.edu (however, you should use your access page for the most reliable responses)
For more general information about the emergency situation, please refer to:
· Web site - www.vhcc.edu
· Telephone Number - 276-739-2400
· Emergency Text Messaging or Phone System- Virginia Highlands Community College uses VHCC Alert to immediately contact you during a major crisis or emergency. VHCC Alert delivers important emergency alerts, notifications and updates to you on your E-mail account (work, home, other), cell phone, pager or smartphone/PDA (BlackBerry, Treo & other handhelds). VHCC Alert is a free service offered by VHCC. Your wireless carrier may charge you a fee to receive messages on your wireless device. VHCC will test the alert system each semester. Register online at alert.vhcc.edu or by sending a text message to 411911 keyword: VHCC
In the event of an emergency just regarding this class, the instructor will contact all students via email, and may post information to your access site. You should check both email and your access site.