# Portland Community College

Course Number:
MTH 111
Course Title:
College Algebra
Credit Hours:
5
Lecture Hours:
50
Lecture/Lab Hours:
0
Lab Hours:
0
Special Fee:

#### Course Description

Explores relations and functions graphically, numerically, symbolically, and verbally. Examines exponential, logarithmic, polynomial, and rational functions. Investigates applications from a variety of perspectives. Graphing calculator required. TI-89 Titanium or Casio ClassPad recommended. The PCC math department recommends that students take MTH courses in consecutive terms. Prerequisites: MTH 95, RD 115, and WR 115, or equivalent placement. Recommended: MTH 95 taken within the past 4 terms. Audit available.

#### Intended Outcomes for the course

Upon successful completion students should be able to:

• Demonstrate an understanding of functions including function notation, function algebra, domain/range, inverse functions, piecewise functions, graph transformations, and symmetry.
• Analyze polynomial, rational, exponential, and logarithmic functions represented numerically, symbolically, verbally and graphically and identify properties of these functions using technology.
• Use variables to represent unknown quantities; create models; solve exponential, logarithmic, polynomial, and rational equations; and interpret the results.
• Demonstrate a mastery of the skills necessary for future course work that requires the use of college algebra concepts.

#### Course Activities and Design

All activities will 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 techniques. Activities will include group work.

#### Outcome Assessment Strategies

Assessment shall include:

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

1. Evaluating logarithmic expressions

2. Solving logarithmic equations

3. Solving exponential equations

4. Function algebra

2. Graphically

1. Graphing Polynomials

2. Graphing Rational functions

3. Transformations of functions

2. At least two proctored, closed-book, no notes 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 60% 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. Take-home examinations

3. Quizzes

4. Group projects

5. In-class activities

6. Portfolios

7. Individual projects

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

#### Course Content (Themes, Concepts, Issues and Skills)

Course Topics

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

1.0  Explore and analyze functions represented in a variety of forms (numerically, symbolically, verbally and graphically).

1.1  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.

1.2  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 and concave up or concave down as appropriate.

1.3  Construct and express understanding of new functions from functions represented in any form.

1.3.1  Evaluate and simplify the difference quotient.

1.3.2 Construct and express understanding of a sum, difference, product or quotient of two given functions.

1.3.3 Construct and express understanding of a composition of two given functions.

1.3.4 Construct and express understanding of the inverse of a given function.

1.3.5 Investigate and express understanding of the new functions in context of applications.

1.4 Investigate families of functions in any form within the context of transformations.

1.4.1 Shift, reflect and/or stretch a given function horizontally or vertically.

1.4.2 Investigate and express understanding of given transformations in context of applications.

1.4.3 Investigate and express understanding of the symmetry of even and odd functions.

2.0  Explore and analyze exponential functions represented in a variety of forms (numerically, symbolically, verbally and graphically) in context of applications.

2.1 Given an exponential function that is represented graphically, numerically or symbolically, express it in the other two forms.

2.2 Write the symbolic form of exponential functions represented in various forms.

2.2.1  Given two points from an exponential function, generate a model symbolically.

2.2.2  Given initial value and growth rate, generate a model symbolically.

2.2.3  Given a table of values, determine if the model is linear or exponential and generate an appropriate model symbolically.

2.2.4  Given the graph of the function, generate a model symbolically.

2.3  Solve exponential equations symbolically, distinguishing between exact and approximate solutions.

2.4  Investigate different forms of exponential functions including the following:

$f\left(t\right)=a{b}^{t}$

$g\left(t\right)=a{e}^{kt}$

$P\left(t\right)={P}_{0}{\left(1+\frac{r}{n}\right)}^{nt}$

$A\left(t\right)=Pe{}^{rt}$

2.5 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.

3.0 Explore and analyze logarithmic functions represented in a variety of forms (numerically, symbolically, verbally and graphically) in context of applications.

3.1 Express logarithmic functions, using a variety of bases in addition to $e$ and 10, as inverse functions of exponential functions represented in various forms.

3.2 Given a logarithmic function that is represented graphically, numerically or symbolically, the student should be able to express it in the other two forms.

3.3 Using properties of logarithms, including change of base, simplify logarithmic expressions and solve logarithmic equations graphically and symbolically, distinguishing between exact and approximate solutions.

3.4 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.

4.0 Explore and analyze polynomial functions represented in a variety of forms (numerically, symbolically, verbally and graphically) in context of applications.

4.1 Investigate the end-behavior of power functions.

4.2 Given a polynomial function that is represented graphically, represent it symbolically

4.3 Given a polynomial function in factored form, graph it by hand.

4.4 Distinguish the relationship between zeros, roots, solutions and the horizontal-intercepts of a polynomial function.

4.5 Find and estimate zeros of a polynomial that is represented in a variety of forms.

4.5.1 Distinguish between exact and approximate solutions, including complex solutions.

4.6 Sketch a polynomial function given the roots of the function, and the corresponding multiplicity of each root.

4.7 Solve a variety of applied problems involving polynomial functions.  All variables in applications shall be appropriately defined with units.

5.0 Explore and analyze rational functions represented in a variety of forms (numerically, symbolically, verbally and graphically) in context of applications.

5.1  Given a rational function that is represented graphically, represent it symbolically.

5.2  Given a rational function in factored form, graph it by hand.

5.3  Find horizontal asymptotes, vertical asymptotes, and holes of rational functions.

5.4  Recognize oblique asymptotes graphically.

5.5  Understand the concept of limits in the context of asymptotes.

5.6  Solve a variety of applied problems involving rational functions. All variables in applications shall be appropriately defined with units.

6.0  Use technology to enhance understanding of concepts in this course.

6.1 Demonstrate the ability to:

6.1.1  Graph functions in an appropriate window.

6.1.2  Use zoom features.

6.1.3  Graphically find max/min values, zeros/roots, and intersection points without using the trace feature.

6.1.4  Evaluate logarithms of various bases.

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.