Portland Community College | Portland, Oregon

Upon successful completion of this course students will be able to:

• Choose and perform accurate intermediate-level algebraic computations in a variety of situations with and without a calculator.
• Creatively and confidently apply intermediate-level algebraic problem solving strategies.
• Be prepared for future course work.

Assessment shall include at least two of the following measures:
1. Tests
2. Attendance
3. Portfolios
4. Individual student conference

Intermediate Algebra (MTH 95)
Themes:
Functions and function notation
Functions represented as graphs, tables, equations and in words
Connection between symbolic and graphical representations
Algebraic simplification of expressions and solving of equations
Problem solving and modeling, interpreting results in practical terms
Language of graphs
Skills:
Calculator (integrated throughout the course)
Use the home screen carry out arithmetic operations
Use the calculator's table feature to explore functions
Graph functions
. Input the appropriate window settings to view the graph
. Use calculation tools
Value
Zero
Maximum
Minimum
ntersect
Understand that the calculator has limitations
Functions
Understand and apply the definition of function
Determine whether one quantity is a function of another algebraically, graphically, numerically and within real life
contexts by applying the definition of a function
Domain
. Understand the definition of domain (the set of all possible inputs)
Determine the domain of functions represented graphically, algebraically, numerically and verbally
Represent the domain in both interval and set notation, where appropriate
Apply unions and intersections (“and” and “or”) when finding and stating the domain of functions
Understand how the context of a function used as a model can limit the domain
Range
. Understand the definition of range (set of all possible outputs)
. Determine the range of functions represented graphically, numerically and verbally
. Represent the range in interval and set notation, where appropriate
Function notation
Evaluate functions with given inputs using function notation where functions are represented graphically, algebraically,
numerically and verbally (e.g. evaluate )
Algebraically simplify and distinguish between different examples such as , ,
and
nterpret in the appropriate context e.g. interpret where models a real-world function
Solve function equations where functions are represented graphically, algebraically, numerically and verbally (i.e.
solve for and solve for where and should include but not be limited to linear
functions, quadratic functions, and absolute value functions)
Solve function inequalities algebraically (i.e. , , and where and are
linear functions and and where is an absolute value function)
Solve function inequalities graphically (i.e. , , and where and should
include but not be limited to linear functions, and for quadratic and absolute value functions)
Graphs of functions
Use the language of graphs and understand how to present answers to questions based on the graph (i.e. read the
value of an intersection to solve an equation and understand that is a number not a point)
Determine function values, solve equations and inequalities, and find domain and range given a graph
Apply function notation to prerequisite skill of finding linear equations given two ordered pairs
Quadratics
Recognize a quadratic equation given in standard form, vertex form and factored form
Solve quadratic equations by completing the square
Find complex solutions to quadratic equations by the quadratic formula or by completing the square
Understand the graphical implications (i.e. when there is a complex number as a solution to a quadratic equation)
nterpret the meaning in the context of an application
Quadratic functions in vertex form
Graph a parabola after obtaining the vertex form of the equation by completing the square
Given a quadratic function in vertex form or as a graph, observe the vertical shift and horizontal shift of the graph
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)
Determine the domain and range of quadratic functions algebraically and graphically
Applications
Understanding in context: given a quadratic function in algebraic or graphical form find and interpret, including units,
the meaning of the:
Vertex as a maximum or minimum
Vertical intercept
Zeroes/horizontal intercepts/roots
nputs and outputs of functions (e.g. and )
Clearly define variables including appropriate units
State conclusions to applied problems in complete sentences including appropriate units
Explore quadratic functions graphically using the graphing calculator. Convey results using function notation. Examine
the following features:
. Vertex
. Vertical intercept
. Horizontal intercepts
Radical Functions
Understand nth roots
Determine the domain of radical functions with both even and odd roots algebraically and graphically
Determine the range graphically
Understand radicals as expressions with rational exponents and vice versa
Use rational exponents to simplify radical expressions (See addendum)
Practice prerequisite skills of exponents rules in the context of rational exponents
Rationalize denominators so students can recognize equivalent expressions (e.g. )
Solve radical equations algebraically and graphically
. Verify solutions algebraically
Understand that extraneous solutions found algebraically do not appear as solutions on the graph
. Solve literal radical equations for a specified variable
Calculator
. Approximate radicals as powers with rational exponents
. Find the domain and range of radical functions
. Solve radical equations graphically
. Use graphical solutions to check the validity of algebraic solutions
Rational Functions
Determine the domain of rational functions algebraically and graphically
Simplify rational functions, understanding that domain conditions lost during simplification MUST be noted
Rewrite rational expressions by
. Canceling factors common to the numerator and denominator
. Multiplying
. Dividing using both and notation
Simplify the following cases where a, b, c, d represent real numbers, linear polynomials or quadratic polynomials:
, and . (See addendum)
. Adding
. Subtracting
. Simplifying complex rational expressions
he following forms of complex rational expressions shall be simplified: , , , and
where , , , , , and represent real numbers, linear polynomials in one variable, or quadratic
polynomials in one variable. (See addendum.)
Solve rational equations
. Check solutions algebraically
Solve literal rational equations for a specified variable
. Introduce variables with subscripts
Applications
. Solve distance, rate and time problems involving rational terms using well defined variables and stating conclusions
in complete sentences including appropriate units
. 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’s limitations include the following: when
finding horizontal intercepts, the calculator sometimes gives something like 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 touches the x axis; the calculator does not show holes on rational function
graphs; the calculator cannot handle very large numbers, e.g. etc.
For dividing rational expressions as in 5.3.3 and 5.3.3.1, focus on examples where the letters represent real
numbers and linear polynomials. E.g. , , and .
Exploration of difficult rational exponents, as in 4.5, 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. , , , .
As much as possible, instructors should present functions that model real-world problems and relationships to
address the content outlined on this CCOG.
In 3.3.1, when solving applications of quadratic equations, a complex solution should be interpreted as the graph
never reaching a particular real world y-value.
For simplifying complex rational expressions as in 5.3.6.1, a major emphasis shall be placed on cases where
, , , , , and (as above) represent real numbers, linear polynomials in one variable. For
example, or would be good examples.