Portland Community College | Portland, Oregon

Lab time will be devoted primarily to small group activities emphasizing conceptual understanding and appropriate technology. The student’s role is to actively engage in positive collaboration with peers. Activities that can be used during lab are on the Mathematics Department home page at /programs/math/course-downloads.html

Upon completion of this course students should be able to:

1. Explore the concept of a function numerically, symbolically, verbally, and graphically and identify properties of functions both with and without technology.

2. Analyze polynomial, rational, exponential, and logarithmic functions, as well as piecewise-defined functions, in both algebraic and graphical contexts, and solve equations involving these function types.

3. Demonstrate algebraic and graphical competence in the use and application of functions including notation, evaluation, domain/range, algebraic operations & composition, inverses, transformations, symmetry, rate of change, extrema, intercepts, asymptotes, and other behavior.

4. Use variables and functions to represent unknown quantities, create models, find solutions, and communicate an interpretation of the results.

5. Determine the reasonableness and implications of mathematical methods, solutions, and approximations in context.

All activities should follow the premise that formal definitions and procedures evolve from the investigation of practical problems. In-class time is primarily activity/discussion emphasizing problem solving and active learning. Activities will include group work.
 

Assessment shall include:

1. The following must be assessed at least once without the use of books, notes, or calculators, in a proctored exam setting:

   1. Algebraically

      1. Evaluating logarithmic expressions

      2. Solving logarithmic equations

      3. Solving exponential equations

      4. Combining functions using the four arithmetic operations as well as function composition

      5. Finding the inverse of a function

   2. Graphically

      1. Graphing Polynomials

      2. Graphing Rational functions

      3. Transformations of functions

2. At least two proctored, closed book, exams, one of which is a comprehensive final exam that is worth at least 25% of the overall grade. The proctored exams should be worth at least 50% of the overall grade. These exams must consist primarily of free response questions although a limited number of multiple choice and/or fill in the blank questions may be used where appropriate.

3. Various opportunities to express – and be graded on – mathematical concepts in writing. Assessment should be made on the basis of using correct mathematical syntax, appropriate use of the English language, and explanation of the mathematical concept.

4. At least two of the following additional measures:

   1. Graded homework

   2. Quizzes

   3. Group projects

   4. Class activities

   5. Portfolios

   6. Individual projects/ written assignments

   7. Discussion Posts

5. Additional forms of assessment that do not have to be part of the grade:

   1. Attendance

   2. Individual student conference

   3. In-class participation

6. The Lab component will account for at least 15% of the grade, which will incorporate laboratory reports (graded work with an emphasis on proper notation and proper documentation) from students.

Course Topics

1. Functions
2. Exponential Functions and Equations
3. Logarithmic Functions and Equations
4. Polynomial Functions
5. Rational Functions
6. Technology

Course Content

1. Explore and analyze functions represented in a variety of forms (numerically, symbolically, verbally, and graphically).
   1. Given a function in any form, identify and express understanding of the domain and range, the horizontal intercept(s), the vertical intercept, the asymptotes as appropriate, and the end behavior.
   2. Given a function represented graphically, identify and express an understanding of the local and absolute extrema and the approximate intervals over which the function is increasing or decreasing as appropriate.
   3. Calculate and interpret the average rate of change of a function over a specified interval.
   4. Construct and express understanding of new functions and their domains from other functions, represented graphically, symbolically, verbally, and numerically.
      1. Evaluate and simplify the difference quotient.
      2. Construct and express understanding of a sum, difference, product, or quotient of two given functions.
      3. Construct and express understanding of a composition of two given functions.
      4. Construct and express understanding of the inverse of a given function, including domain and range.
      5. Construct and express understanding of piecewise functions.
      6. Investigate and express understanding of the new functions in context of applications.
   5. Investigate families of functions in any form within the context of transformations.
      1. Shift, reflect and/or stretch a given function horizontally or vertically.
      2. Investigate transformations in factored and non-factored forms, e.g., \( f(-2x-8)=f(-2(x+4)) \).
      3. Determine the domain and range of a transformed function.
      4. Investigate and express understanding of given transformations in context of applications.
      5. Investigate and express understanding of the symmetry of even and odd functions from a graphical and algebraic perspective.
2. Explore and analyze exponential functions represented in a variety of forms (numerically, symbolically, verbally and graphically) in context of applications.
   1. Given an exponential function that is represented graphically, numerically or symbolically, express it in the other two forms.
   2. Find the algebraic form of exponential functions represented in various forms.
      1. Given two points from an exponential function, find an algebraic formula for the function.
      2. Given an initial value and growth rate, generate a symbolic model.
      3. Given a table of values, determine if the data are linear or exponential and generate an appropriate symbolic model.
      4. Given the graph of the function, find an algebraic formula for the function.
   3. Solve exponential equations symbolically and graphically, distinguishing between exact and approximate solutions.
   4. Investigate different forms of exponential functions including the following: \( f(t)=ab^{t} \), \( g(t)=ae^{kt} \), \( P(t)=P_0(1+\frac{r}{n})^{nt} \), and \( A(t)=Pe^{rt} \).
   5. Model and solve a variety of applied problems involving exponential functions (such as radioactive decay, bacteria growth, population growth, and compound interest). All variables in applications shall be appropriately defined with units and results should be interpreted in context.
3. Explore and analyze logarithmic functions represented in a variety of forms (numerically, symbolically, verbally, and graphically) in context of applications.
   1. Express logarithmic functions, using a variety of bases in addition to \( e \) and \( 10 \), as inverse functions of exponential functions represented in various forms.
   2. Given a logarithmic function that is represented graphically, numerically, or symbolically, the student should be able to express it in the other two forms.
   3. Using properties of logarithms (including change of base), simplify logarithmic expressions and solve logarithmic equations graphically and symbolically, distinguishing between exact and approximate solutions and recognizing extraneous solutions. 
   4. Solve a variety of applied problems involving logarithmic functions (such as intensity of sound, earthquake intensity, and determining acidity of a solution by its pH). All variables in applications shall be appropriately defined with units and results should be interpreted in context.
4. Explore and analyze polynomial functions represented in a variety of forms (numerically, symbolically, verbally, and graphically) in context of applications.
   1. Investigate the end-behavior of power functions.
   2. Given a polynomial function that is represented graphically, find a symbolic representation.
   3. Given a polynomial function in factored form, graph it by hand.
   4. Distinguish the relationship between zeros, roots, solutions, and the horizontal-intercepts of a polynomial function.
   5. Find and estimate zeros of a polynomial that is represented in a variety of forms.
      1. Distinguish between exact and approximate solutions.
   6. Sketch a graph of a polynomial function given the roots of the function and the corresponding multiplicity of each root.
   7. Solve a variety of applied problems involving polynomial functions. All variables in applications shall be appropriately defined with units and results should be interpreted in context.
5. Explore and analyze rational functions represented in a variety of forms (numerically, symbolically, verbally, and graphically) in context of applications.
   1. Given a rational function that is represented graphically, represent it symbolically.
   2. Given a rational function in factored form, graph it by hand.
   3. Given a rational function represented symbolically:
      1. Analyze the long-run behavior by using the ratio of leading terms:
         • Finding horizontal asymptotes.
         • Recognizing when here are no horizontal asymptotes.
      2. Find vertical asymptotes.
      3. Find any holes.
   4. Solve a variety of applied problems involving rational functions. All variables in applications shall be appropriately defined with units and results should be interpreted in context.
6. Use technology to enhance understanding of concepts in this course.
   1. Demonstrate the ability to:
      1. Graph functions in an appropriate viewing screen.
      2. Graphically find max/min values, zeros/roots, and intersection points.

ADDENDUM

Documentation Standards for Mathematics: All work in this course will be evaluated for your ability to meet the following writing objectives as well as for "mathematical content."

1. Every solution must be written in such a way that the question that was asked is clear simply by reading the submitted solution.
2. Any table or graph that appears in the original problem must also appear somewhere in your solution.
3. All graphs that appear in your solution must contain axis names and scales. All graphs must be accompanied by a figure number and caption. When the graph is referenced in your written work, the reference must be by figure number. Additionally, graphs for applied problems must have units on each axis and the explicit meaning of each axis must be self-apparent either by the axis names or by the figure caption.
4. All tables that appear in your solution must have well defined column headings as well as an assigned table number accompanied by a brief caption (description). When the table is referenced in your written work, the reference must be by table number.
5. A brief introduction to the problem is almost always appropriate.
6. In applied problems, all variables and constants must be defined.
7. If you used the graph or table feature of your calculator in the problem solving process, you must include the graph or table in your written solution.
8. If you used some other non-trivial feature of your calculator (e.g. SOLVER), you must state this in your solution.
9. All (relevant) information given in the problem must be stated somewhere in your solution.
10. A sentence that orients the reader to the purpose of the mathematics should usually precede symbol pushing.
11. Your conclusion shall not be encased in a box, but rather stated at the end of your solution in complete sentence form.
12. Line up your equal signs vertically.
13. If work is word-processed all mathematical symbols must be generated with a math equation editor.