Portland Community College | Portland, Oregon

Effective Fall 2017, students will no longer be required to have physical graphing calculators in either MTH 95 or MTH 111. Where physically possible instructors will demonstrate using Desmos, GeoGebra, or other online programs in class. Assessments requiring the use of a graphing calculator will be done outside of proctored exams.

There is a required notation addendum and required exercise supplement for this course which may be found at pcc.edu/programs/math/course-downloads.html.

Upon completion of the course students should be able to:

• Manipulate absolute value, quadratic, rational and radical expressions, equations, and/or inequalities.
• Translate between graphical, numerical, verbal, and symbolic representations of functions.
• Use variables to represent unknown quantities, create a function to model a situation and use algebra and/or technology to find and interpret a result.
• Interpret properties of functions and relations, such as the meaning of ordered pairs, domain and range, maximum and minimum values, and intercepts. 

• Develop algebraic skills necessary to accurately simplify, evaluate, and/or interpret properties of absolute value, quadratic, rational and radical
  expressions and/or equations.
• Participate in activities, assignments, and exams, which show an understanding of the connection between graphical, numerical, verbal, and symbolic
  representations of functions.
• Participate in activities, assignments, and exams which apply algebraic functions to real-life situations while showing an understanding of the creation and interpretation of a function to model the situation.
• Participate in, and contribute to, class discussions and activities.
• Take all scheduled examinations.

Assessment Requirements

1. The following must be assessed in a proctored, closed-book, no-note, and no-calculator setting:
   • finding the equation of the linear function given two ordered pairs stated using function notation
   • simplifying rational expressions
   • solving rational equations
   • solving radical equations
   • graphing quadratic functions
   • determining the domain of radical and rational functions
   • evaluating algebraic expressions that include function notation
2. At least two proctored, closed-book, no-note examinations (one of which is the comprehensive final) must be given. 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. Assessment must include evaluation of the students ability to arrive at correct conclusions using proper mathematical procedures and notation. Additionally, each student must be assessed on their ability to use appropriate organizational strategies and write appropriate conclusions. Application problems must be answered in complete sentences.
4. At least two of the following additional measures must also be used
   1. Take-home examinations
   2. Graded homework
   3. Quizzes
   4. Projects
   5. In-class activities
   6. Portfolios

Themes:

• Functions represented as graphs, tables, equations and in words, and connections among these representations
• Algebraic manipulation
• Graphical understanding
• Problem solving
• Effective communication
• Critical thinking
• Applications, formulas, and modeling
• Use of Technology

Skills:

1. Calculator (integrated throughout the course)
   1. Use the home screen to carry out arithmetic operations
   2. Use the calculator's table feature to explore functions
   3. Graph functions
      1. Input the appropriate window settings to view the graph
      2. Use calculation tools
         1. Value
         2. Zero
         3. Maximum
         4. Minimum
         5. Intersect
   4. Understand that the calculator has limitations
2. Functions
   1. Understand and apply the definition of function
   2. Determine whether one quantity is a function of another algebraically, graphically, numerically and within real-world contexts by applying the definition of a function
   3. Domain
      1. Understand the definition of domain (the set of all possible inputs)
      2. Determine the domain of functions represented graphically, algebraically, numerically and verbally
      3. Represent the domain in both interval and set notation, where appropriate
      4. Apply unions and intersections (OR and AND) when finding and stating the domain of functions
      5. Understand how the context of a function used as a model can limit the domain
   4. Range
      1. Understand the definition of range (the set of all possible outputs)
      2. Determine the range of functions represented graphically, numerically and verbally
      3. Represent the range in interval and set notation, where appropriate
   5. Function notation
      1. Evaluate functions with given inputs using function notation where functions are represented graphically, algebraically, numerically and verbally (e.g. evaluate \(f(7)\))
      2. Algebraically simplify and distinguish between different examples such as \(f(x + 2)\), \(f(x) + 2\), \(3f(x)\), and \(f(3x)\)
      3. Interpret \(f(a) = b\) in the appropriate context e.g. interpret \(f(3) = 5\) where \(f\) models a real-world function
      4. Solve function equations where functions are represented graphically, algebraically, numerically and verbally (i.e. solve \(f(x) = b\) for \(x\) and solve \(f(x) = g(x)\) for \(x\) where \(f\) and \(g\) should include but not be limited to linear functions, quadratic functions, and absolute value functions)
      5. Solve function inequalities algebraically (i.e. \(f(x) \gt b\), \(f(x) \gt g(x)\), and \(a \lt f(x) \lt b\) where \(f\) and \(g\) are linear functions and \(f(x) \gt b\) and \(f(x) \lt b\) where \(f\) is an absolute value function)
      6. Solve function inequalities graphically (i.e. \(f(x) \gt b\), \(f(x) \gt g(x)\), and \(a \lt f(x) \lt b\) where \(f\) and \(g\) should include but not be limited to linear functions, and \(f(x) \gt b\) for quadratic and absolute value functions)
   6. Graphs of functions
      1. Use the language of graphs and understand how to present answers to questions based on the graph (i.e. read the \(x\)-value of an intersection to solve an equation and understand that \(f(2)\) is a number not a point)
      2. Determine function values, solve equations and inequalities, and find domain and range given a graph
   7. Apply function notation to prerequisite skill of finding linear equations given two ordered pairs
3. Complex Numbers
   1. Perform operations using complex numbers (i.e. add, subtract, multiply, and divide)
   2. Rationalize complex denominators (e.g. \(\frac{2}{3+i}\), \(\frac{5}{i}\))
4. Absolute Value
   1. Solve absolute value equations and inequalities
      1. Solve absolute value function equations where functions are represented graphically, algebraically, numerically and verbally (e.g. solve \(f(x) = b\) for \(x\) and solve \(f(x) = g(x)\))
      2. Solve absolute value function inequalities algebraically and graphically (e.g. \(f(x) \gt b\), \(f(x) \lt b\))
5. Quadratics
   1. Recognize a quadratic equation given in standard form, vertex form and factored form
   2. Solve quadratic equations by completing the square
   3. Find complex solutions to quadratic equations by the quadratic formula or by completing the square
      1. Understand the graphical implications (e.g. when there is a complex number as a solution to a quadratic equation)
      2. Interpret the meaning in the context of an application
   4. Quadratic functions in vertex form
      1. Graph a parabola after obtaining the vertex form of the equation by completing the square
      2. Given a quadratic function in vertex form or as a graph, observe the vertical shift and horizontal shift of the graph \(y = x^2\)
      3. Connect graphing via vertex form with the prerequisite graphing methods (i.e. axis of symmetry, horizontal intercepts, vertex formula, vertical intercept, points found by symmetry)
   5. Determine the domain and range of quadratic functions algebraically and graphically
   6. Applications
      1. Understanding in context: given a quadratic function in algebraic or graphical form find and interpret, including units, the meaning of the:
         1. Vertex as a maximum or minimum
         2. Vertical intercept
         3. Zeros/horizontal intercepts/roots
         4. Inputs and outputs of functions (e.g. \(f(x) = 5\) and \(f(2)\))
      2. Clearly define variables including appropriate units
      3. State conclusions to applied problems in complete sentences including appropriate units
   7. Explore quadratic functions graphically using the graphing calculator. Convey results using function notation. Examine the following features:
      1. Vertex
      2. Vertical intercept
      3. Horizontal intercepts
6. Radicals
   1. Understand \(n^{\mathrm{th}}\) roots
   2. Determine the domain of radical functions with both even and odd roots algebraically and graphically
   3. Determine the range graphically
   4. Understand radicals as equivalent to expressions with rational exponents and vice versa
   5. Use rational exponents to simplify radical expressions (See addendum)
   6. Practice prerequisite skills of exponents rules in the context of rational exponents
   7. Rationalize denominators so students can recognize equivalent expressions (e.g. \(\frac{4}{2-\sqrt{5}}\), \(\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}\)
   8. Solve radical equations algebraically and graphically
      1. Verify solutions algebraically
      2. Understand that extraneous solutions found algebraically do not appear as solutions on the graph
      3. Solve literal radical equations for a specified variable
   9. Calculator
      1. Approximate radicals as powers with rational exponents
      2. Find the domain and range of radical functions
      3. Solve radical equations graphically
      4. Use graphical solutions to check the validity of algebraic solutions
7. Rational Functions
   1. Determine the domain of rational functions algebraically and graphically
   2. Simplify rational functions, understanding that domain conditions lost during simplification MUST be noted
   3. Rewrite rational expressions by
      1. Canceling factors common to the numerator and denominator
      2. Multiplying
      3. Dividing using both \(\frac{\frac{a}{b}}{\frac{c}{d}}\) and \(\frac{a}{b}\div \frac{c}{d}\)
         1. Simplify the following cases where \(a\), \(b\), \(c\), and \(d\) represent real numbers, linear polynomials or quadratic polynomials: \(\frac{a}{\frac{b}{c}}\), \(\frac{\frac{a}{b}}{c}\), and \(\frac{\frac{a}{b}}{\frac{c}{d}}\). (See addendum)
      4. Adding
      5. Subtracting
      6. Simplifying complex rational expressions
         1. The following forms of complex rational expressions shall be simplified: \(\frac{a}{\frac{b}{c}+\frac{d}{e}}\), \(\frac{\frac{b}{c}+\frac{d}{e}}{a}\), \(\frac{\frac{a}{b}}{\frac{c}{d}+\frac{e}{f}}\), and \(\frac{\frac{a}{b}+\frac{c}{d}}{\frac{e}{f}}\), where \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\) represent real numbers, linear polynomials in one variable, or quadratic polynomials in one variable. (See addendum)
   4. Solve rational equations
      1. Check solutions algebraically
   5. Solve literal rational equations for a specified variable
      1. Introduce variables with subscripts
   6. Applications
      1. Solve distance, rate and time problems involving rational terms using well defined variables and stating conclusions in complete sentences including appropriate units
      2. Solve problems involving work rates using well defined variables and stating conclusions in complete sentences including appropriate units

ADDENDUM:

Functions should be studied symbolically, graphically, numerically and verbally.

As much as possible, instructors should present functions that model real-world problems and relationships to address the content outlined on this CCOG.

Function notation is emphasized and should be used whenever it is appropriate in the course.

Students should be required to use proper mathematical language and notation. This includes using equal signs appropriately, labeling and scaling the axes of graphs appropriately, using correct units throughout the problem solving process, conveying answers in complete sentences when appropriate, and in general, using the required symbols correctly.

Students should understand the fundamental differences between expressions and equations including their definitions and proper notations.

All mathematical work should be organized so that it is clear and obvious what techniques the student employed to find his answer. Showing scratch work in the middle of a problem is not acceptable.

Since technology is used throughout the course, there is a required calculator packet for students that gives directions for several graphing calculators. The students should understand the limitations of calculator (i.e. when the calculator gives misleading information). Examples of the calculator limitations include the following: when ending horizontal intercepts, the calculator sometimes gives something like \(\mathtt{y=3E{-}13}\); the calculator rounds to 12 or fewer decimal places; some calculators appear to show vertical asymptotes on the graphs of rational functions; it appears that the graph of \(y = \frac{1}{x^2}\) touches the \(x\)-axis; the calculator does not show holes on rational function graphs; the calculator cannot handle very large numbers (e.g. \(10^{1000}\)).

For dividing rational expressions, focus on examples where the letters represent real numbers and linear polynomials. E.g. \(\frac{\frac{x^2-4}{x^2+x-2}}{x-2}\), \(\frac{x^2-9}{\frac{3x+5}{4x+10}}\), and \(\frac{\frac{x^2-9}{x^2-x-12}}{\frac{6+2x}{x^2-7x+12}}\).

Exploration of difficult rational exponents should be limited. Basic understanding is essential and a deep understanding takes more than one course to develop. Examples should be limited to one or two variables, keeping things as simple as possible while covering all possibilities. E.g. \(5x^{1/2}\), \(3x^{1/3}\), \(\frac{2x^{1/3}}{x^{1/2}}\), and \(4y^{1/2}y^{1/3}\).

When solving applications of quadratic equations, a complex solution should be interpreted as the graph never reaching a particular real world output value.

For simplifying complex rational expressions, a major emphasis shall be placed on cases where \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\) (as above) represent real numbers, linear polynomials in one variable. For example, \(\frac{\frac{1}{x}+\frac{1}{x+2}}{\frac{5}{x+1}}\) and \(\frac{3}{\frac{1}{x}+\frac{5}{x^2}}\) would be good examples.