Course Title and Description: Mth 165, Precalculus. (4 credits): Presents college algebra, matrices, and algebraic, exponential and logarithmic functions and trigonometric functions.  Prerequisite:  Algebra I, Algebra II and Geometry.

The broad goals 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 linear, quadratic, exponential, logarithmic, power, polynomial functions and trigonometric functions as well as recurrence relations and matrices.

Each 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.  This list is not exhaustive, but covers at least 80% of the topics for which students will be responsible:

Conceptual Objectives

Describe the behavior of a given table or graph in terms of increasing and decreasing behavior, asymptotes, cyclical behavior, and the increasing, decreasing or cyclical behaviors of the rates of increase or decrease.

Explain the relationships among the formula, the standard table (using the standard graph points for each), and the graph for each basic family of functions over its standard domain. These families include the fundamental power functions (p=-2, -1, .5, 1, 2, 3 and 4), linear functions, quadratic functions and exponential functions.

Explain the relative shapes of the graphs of the fundamental power functions, with attention to how these shapes change from one power to the next.

Describe the behavior of each basic function and its transformations (in terms of the behaviors outlined in #1), and explain how each behavior is related to the formula for the function.

Explain how to use the quadratic formula to graph a given quadratic function.

Give examples of how behavior which can be modeled with each of the basic functions can arise in the real world.

List and explain in commonsense terms the fundamental laws of real numbers, and show how each can be useful in understanding the real world.

Explain how the sequence behavior of each of the basic functions is related to the recurrence relation from which the corresponding sequence might arise. For each function type describe a real-world situation in which the sequence behavior corresponds to expected real-world behavior, and how the recurrence relation is related to that real-world behavior.

Explain how the sequence behavior of each of the given functions is related to the difference equation from which the corresponding sequence might arise. For each function type describe a real-world situation in which the sequence behavior corresponds to expected real-world behavior, how the recurrence relation is related to that real-world behavior, and how the difference equation is related to that real-world behavior.

For each basic function f(x), describe how each of the parameters A, k, c and h affects the graph of y = A f(k(x-h))+c, and describe how to construct for any function f(x) the graph of y = A f(k(x-h))+c from the graph of y = f(x).

Explain geometrically why the family of four-parameter transformations A f(k(x-h)) + c can be achieved by varying only two parameters if f(x) is a linear function, while three parameters are required for quadratic, exponential and power functions (with specified power), and four parameters are required for arbitrary power (power not specified) and trigonometric functions.

Explain how to construct a table of fluid amount vs. time from a table of flow rate vs. time, and a table of flow rate vs. time from a graph of fluid amount vs. time. Explain how to generalize to the task of finding an amount chronicle from a rate chronicle and a rate chronicle from an amount chronicle.

Explain how to construct a graph of fluid amount vs. time from a graph of flow rate vs. time, and a graph of flow rate vs. time from a graph of fluid amount vs. time. Explain how to generalize to the task of finding an amount graph from a rate graph and a rate graph from an amount graph.

Explain how to use trapezoids or fundamental triangles to determine the average rate at which some function changes over a specified time interval, or to determine the approximate change in amount over a specified time interval due to a given rate function.

Explain why the rate chronicle obtained from an amount chronicle obtained from a rate chronicle should be close to that original rate chronicle, provided the time intervals are sufficiently small. Explain how the graph of f(x) can be used to estimate the solution(s) of the equation f(x) = 0, f(x) = c, and how the graphs of f(x) and g(x) can be used to estimate the solution(s) of the equation f(x) = g(x).

Describe the rate-of-change behavior associated with linear, quadratic, power or exponential functions.

Describe situations in which linear, quadratic, power or exponential functions arise naturally.

Explain the connection between the rate-of-change behavior of a linear, quadratic or exponential function and its formula or its graph.

Explain how to use fundamental triangles to determine the equation of a line in the plane through two given points or of a circle with given center and radius.

Give examples of real-world in which simultaneous equations in two or more variables arise.

Give examples of how systems of simultaneous equations permit us to fit a linear function to two data points, a power, quadratic or exponential function to three data points, or a polynomial function to four or more points.

Explain how the process of elimination permits us to solve systems of linear equations, and how this process is represented by the process of matrix reduction.

Explain how to optimize the value of a given function f(x,y) over a region of the plane defined by a set of simultaneous linear equations in x and y.

Explain why we can fit a linear function to any two data points and a quadratic, exponential or specified power function to any three.

Explain how any equation can be arranged in the form f(x) = 0, and explain what to look for on the graph of f(x) to determine the solutions.

Give examples of real-world situations in which equations of the type f(x) = 0, f(x) = c or f(x) = g(x) arise.

Explain how to use the graph of a function f(x) to determine its approximate maximum.

Explain how to construct a table or graph of the reciprocal of a function, given a table or graph of the function.

Explain how, given tables or graphs of two functions, to construct the graph of the sum, the product, the difference, the quotient or (either) composite of the two functions.

Describe real-world situations naturally modeled by a function of a function, and explicitly describe the nature of the functions as well as their composite.

Explain how to use the table or the graph of a function to determine whether an inverse function exists and, if so, to construct the table or graph of that inverse function.

Explain how the concept of inverse functions can be used to solve an equation of the form f(x) = c, where f(x) is a function of x, c is a constant and solution means obtaining a value of x which makes the equation true.

Explain how logarithms arise naturally in the attempt to solve exponential equations.

Explain how logarithms are used to model and represent rapid growth in a way that can be grasped intuitively.

Explain how to linearize exponential or power-function data, then use the inverse transformation on the linear regression equation to obtain a functional model.

Explain how the function which linearizes a set of data is related to the function that models the data.

Explain ircular models and the graphs of trigonometric functions

Explain and apply triangle trigonometry and trigonometric identities

Use systems of equations and matrices to model real-world problems

Mechanics and Manipulations

Given a linear, quadratic, polynomial, exponential or logarithmic equation, or any equation involving the basic functions, use appropriate numerical, algebraic, computerized algebraic or graphical techniques to find the solution(s) of the equation.

Algebraically determine the simplest form of the inverse of a given function

Show algebraically why the family of four-parameter transformations A f(k(x-h)) + c can be achieved by varying only two parameters if f(x) is a linear function, while three parameters are required for quadratic, exponential and specified-power functions, and four parameters are required for arbitrary-power and trigonometric functions.

Given a function, a range of values and an increment, construct the corresponding sequence of function values and construct a table for the function.

Using either the sequence and increment or the table, as specified, construct a corresponding table of approximate rates of change. Using either the sequence and increment or the table, as specified, and assuming that the function represents a rate of change with respect to its variable, construct a corresponding table of amount changes cumulative to the initial value of the variable.

Use DERIVE to author, simplify, graph and solve, as appropriate, a given algebraic expression, equation or function.

Use a spreadsheet to compute approximate rates of change and changes in amounts for given amount or rate functions, respectively, given a range of values of the variable and an increment, and to graph the results.

Use DERIVE to construct the graph of a family of functions obtained by changing one or more of the parameters A, k, h and c of the function y = A f(k(x-c))+h, where f(x) is a given function.

Modeling and Problem Solving

Given a situation which can be modeled by one or more algebraically defined functions, give a specific explanatory interpretation of the graph, its slope and/or area (as appropriate to the situation) over a given range of values, and any asymptotes. Show how to represent these quantities algebraically.

Given a situation in which the rate at which a quantity changes depends on that quantity and/or on some independent variable, write the rate-of-change equation that models the situation and use DERIVE to determine the function that solves the equation.

Given a set of dependent variable vs. independent variable data describing a real-world situation, fit an appropriate function to the data. Pose and answer the standard questions related to the behavior of the model (standard questions include determining the value of the independent variable from the dependent, or the dependent from the independent, and rate of change or accumulated amount questions).

Given an unfamiliar problem, use documented graphical and other imaging techniques, algebraic reasoning, geometric reasoning and intuition, and verbal reasoning to attempt a solution.

All assignments will be available on the homepage.   Class notes will be distributed in CD-ROM format.  Test problems, notation and the overall core of the course will be based on the materials on the homepage.  Some assignments refer to the suggested text, which is Precalculus by Sullivan, published by Prentice-Hall.  It is is noted that most students do not use the text, so there is no recommendation to purchase it unless the student feels that it is necessary to do so.

The student will be required to purchasea set of CD-R disks containing video versions of lectures and other materials.

Topics will include:

MTH 165 will cover:

• Modeling of real-world phenomena by use of graphs, functions, equations and computerized models.

• The algebra of functions

The homepage units to be covered (see homepage) include:

Homepage units 1.1-1.8: The Mathematical Modeling process, Modeling by Quadratic Functions, Basic Function Families, Introduction to Rates, and use of DERIVE

Homepage units 2.1-2.7: Modeling by Linear Functions, more DERIVE, Modeling with Proportionality and Power Functions

Homepage units 3.1-3.6: Modeling by Exponential Functions, Inverse Functions and Logarithms, Linearize Data and Curve Fitting

Homepage units 4.1-4.8: Polynomials, Properties of Functions, Combining Functions

Homepage units 5.1-5.7: Sex and Drugs (difference equations related to population patterns and antibiotic retention), DERIVE strategies for curve fitting, Experiments

These section numbers may be modified slightly as materials are revised in response to the needs of the class.

The course will run parallel with the material in Chapters 1-5 of the recommended text.

Students will complete and submit the assignments specified on the homepage.

The instructor will meet with students regularly throughout the class session to assess progress and pose questions.

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Students may on occasion be asked to critique work done by other students. The instructor will not make reference to the identity of any party in this exchange, permitting students to protect their anonymity.

Class notes with links to MPEG video files will be provided via CD-ROM media.    MPEG files can be played by a computer with an MPEG player installed; this software can be purchased or downloaded from the Internet, and will be available on some public-access terminals at VHCC.

A major Quiz, two tests and a final exam will be administered.  The final examination will given the same weight as a regular test; however, if it is to the advantage of the student this final examination will be given double the weight of a regular test.

Assigned work and/or daily quizzes will be graded, based on the student's final mastery of the assignment as evidenced by the initial attempt and followup work based on the critiques received by the student. The average of grades assigned on this work will count as 1/2 of a test grade. If this average is higher than the average on other tests, it will be counted as a full 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: