Math 120 --- Intermediate Algebra
Catalog Description
A course in which functions are investigated graphically, numerically, symbolically and verbally in real-world settings. Linear, quadratic, absolute value, polynomial, rational, radical, exponential, and logarithmic equations and functions are explored. Technology is integrated into all aspects of the course.
Special notes or advisories: A graphing calculator is required.
Prerequisites:
Math 105 or 106 (or equivalent) with a grade of "C" or better or appropriate score on the math placement exam.
Describe representative skills without which the student would be highly unlikely to succeed: Ability to use the properties of real numbers to solve linear equations and inequalities, draw, read, and interpret graphs, and find the equations of lines. Ability to correctly manipulate polynomial expressions, including factoring. Students must possess the ability to use the graphing calculator.
Course Learning Outcomes:
What should the student be able to do as a result of taking this course? State some of the objectives in terms of specific, measurable student accomplishments.
- Students should be able to read, write, and speak accurately about mathematical ideas using correct mathematical notation.
- Students should be able to apply the mathematics they have learned to real-world problems and applications.
- Students should be able to use graphs and the graphing calculator to explore mathematical concepts and to verify their work.
- Students should be able to demonstrate competency in the required prerequisite skills for all transfer level math courses.
- Students should be able to demonstrate the characteristics of an effective learner, such as note-taking, critical reading, etc.
- Students should be able to explain the concept of function, identify the characteristics of different classes of functions, and use functions to solve problems in mathematics.
- Students should be able to demonstrate the algebraic skills that will support success in the other outcomes.
Course Content
- Themes: What themes, if any, are threaded throughout the learning experiences in this course?
- Functions
- Critical thinking
- Problem solving
- Algebraic skills
- Technology
- Graphing and data analysis
- Communication
- Concepts: What concepts do students need to understand to demonstrate course outcomes?
- Setting up models involving real-world applications, solving the models, and verifying the results.
- The presentation of mathematical solutions in a logical coherent structure, including the use of fundamental writing skills, grammar, and punctuation.
- Use of the graphing calculator as a fundamental problem-solving tool.
- The connection between graphs and properties of functions.
- Functions: definition, graphs, properties, and their application in the problem-solving process.
- The fundamental relationship between a function and its inverse, particularly the relationship between the exponential and logarithmic functions.
- The recognition that proper use of algebraic skills is an important tool in multiple problem-solving situations.
- Issues: What primary issues or problems, if any, must students understand to achieve course outcomes.
- The appropriate use of technology in the problem-solving process.
- The role of the student in becoming a successful learner.
- The importance of writing mathematics using correct notation and grammar.
- The connection between mathematics and the real world.
- The recognition that the problem-solving skills learned in this class are applicable in future mathematics classes and classes in related fields, such as physics, engineering, etc.
- The differences between solving an equation and simplifying an expression.
- Skills: What skills must students master to demonstrate course outcomes?
- Use a graphing calculator to: graph a function, adjust the viewing window, trace, find intersections, zeros, and extrema; generate a table; enter data and calculate regression curves; approximate solutions to equations and inequalities.
- Be able to identify the properties of linear, quadratic, polynomial, rational, radical, exponential, logarithmic, and absolute value functions from their graphs, including domain, range, intercepts, extrema, and asymptotic behavior.
- Preliminaries:
- Describe the various number systems.
- Solve linear equations and inequalities.
- Correctly use the conjunctions "and" and "or" in mathematical statements.
- Correctly describe solution sets using both set-builder and interval notation.
- Solve compound linear inequalities.
- Linear functions:
- Identify slope from the graph, and calculate slope from the formula.
- Compute slope as a rate of change and make the connection to real-world applications, using correct units.
- Use slope and intercepts to draw the graph of the function.
- Find the equation of a line in slope-intercept, point-slope, and standard form.
- Find the equations and draw the graphs of parallel and perpendicular lines.
- Using real data, determine a linear model of best fit, and use the model to make predictions, both analytically and graphically.
- Absolute Value Functions:
- Draw the graphs of constant and piecewise-defined functions.
- Determine a piecewise definition for a given absolute value function.
- Solve equations and inequalities involving absolute value, both graphically and analytically.
- Quadratic Functions:
- Given a quadratic function in vertex form, determine the domain, range, vertex, and axis of symmetry, and then use this information to draw the graph.
- Use the technique of completing the square to transform a general quadratic function into vertex form.
- Determine the zeros of a quadratic function using factoring and the quadratic formula.
- Solve quadratic equations.
- Use the equations of motion with constant acceleration to solve applications such as free-fall, projectile motion, braking distance, etc.
- Find maxima and minima of quadratic functions and apply this optimization technique to real-world applications.
- Polynomial Functions:
- Determine the zeros and end behavior of a given polynomial function, and then use this information to sketch its graph.
- Solve polynomial equations using factoring.
- Use the graphing calculator to identify the local extrema of a given polynomial function, and apply this optimization technique to real-world applications.
- Rational Functions:
- Determine the domain, vertical and horizontal asymptotes, removable singularities, and zeros of a given rational function, and then use this information to sketch its graph.
- Add, subtract, multiply, and divide rational expressions, and then reduce the result to lowest terms.
- Simplify complex rational expressions.
- Solve rational equations.
- Solve application problems involving motion and work.
- Exponential and Logarithmic Functions:
- Compute roots using radical notation and rational exponents. Use the laws of exponents to simplify expressions involving rational exponents.
- Determine the domain, range, and horizontal asymptote of an exponential function, and identify whether it exhibits growth or decay. Use this information to sketch its graph.
- Solve finance problems involving discrete and continuous compound interest.
- Determine if a function is one-to-one. If so, compute the formula for its inverse function and sketch its graph.
- Compute the composition of two functions, and use this skill to verify that two given functions are inverses of each other.
- Convert equations between exponential and logarithmic form, and use this skill to compute values of logarithmic functions.
- Determine the domain, range, and vertical asymptote of a logarithmic function.
- Use the change of base formula and a calculator to determine the values of logarithmic functions to different bases.
- Use logarithms to approximate the solutions of exponential equations.
- Solve real-world application problems involving compound interest, population growth, and radioactive decay.
- Radical Functions:
- Determine the domain and range of the square root function. Use this information along with the theory of transformations (scaling, reflections, translations) to graph a variety of square root functions.
- Simplify radical expressions.
- Solve equations involving radical expressions.
- Solve application problems involving the Pythagorean Theorem and the distance formula.
Representative Learning Activities
What will the students be doing (i.e., Listening to lectures, participating in discussions and/or group activities, attending a field trip, etc.)? Relate the activities directly to the Course Learning Outcomes.
- Listening to lectures.
- Participating in group activities and/or assignments.
- Participating in class assignments and/or discussions.
- Completing homework assignments.
- Completing online activities on the computer.
- Using the graphing calculator to complete activities designed to foster a deeper level of understanding of the concepts and skills developed in this class.
Assessment Tasks
How will the student show evidence of achieving the Course Learning Outcomes? Indicate which assessments (if any) are required for all sections.
- Representative assessment tasks:
- In-class examinations and/or quizzes.
- Homework assignments.
- Take-home examinations and/or quizzes allow the instructor to include questions and/or exercises that require the use of the graphing calculator to supplement the analysis. Extra time allows the students to develop their writing and presentation skills.
- Writing assignments designed to develop communication of mathematical concepts.
- Group or individual in-class activities.
- Portfolios and/or reference books.
- Group and/or individual projects and presentations.
- Required assessments for all sections – to include but not limited to:
- Homework assignments.
- At least two proctored, closed-book examinations, plus a comprehensive final examination.
Examples of Appropriate Texts or Other Readings
- Author: Department of Mathematics, College of the Redwoods; Title: Intermediate Algebra, Third Edition; Date 2007