Topics include algebraic, graphical and numerical representations of functions, including composition and inverses of functions. Focuses on the study of rational functions, exponential and logarithmic functions, and trigonometric functions of real numbers and angles. Also includes analytic trigonometry and solutions of triangles. Techniques of problem solving and applications are included throughout the course requiring the frequent usage of graphing calculators. Requires a graphing calculator, with the TI-84 Plus series recommended. Also requires access to an online homework management system.
Goals, Topics, and Objectives
- To develop an understanding of functions represented algebraically, graphically and numerically necessary for the study of higher level science and mathematics.
- To develop an understanding of rational functions, exponential and logarithmic functions and trigonometric functions of real numbers and angles.
- To develop familiarity with some mathematical and physical applications of advanced algebra and trigonometry.
- To incorporate graphing calculators whenever appropriate to illustrate concepts and solve problems.
- To develop in students the problem-solving skills needed to interpret, analyze and solve applied problems requiring precalculus-level skills in advanced algebra and trigonometry.
In the pre-requisite course, students demonstrate knowledge of polynomial functions and the fundamental theorem of algebra.
Upon successful completion of this course, students should be able to:
- Functions and Their Graphs
- Determine the local minima and local maxima of a function and the interval(s) on which the function is increasing, decreasing and/or constant, given the graph of the function.
- Form the sum, difference, product, quotient, and composition of two functions and determine their domains, given their equations or graphs.
- Relate the difference quotient and average rate of change of a function.
- Graph piecewise-defined functions and determine their domains and ranges.
- With the aid of a graphing utility, determine points of maximization or minimization in situations dealing with distances, areas, and volumes of various geometric figures, as well as in real-world business situations involving revenues, costs, and profits.
- Rational Functions
- Find the domain of a rational function.
- Find vertical and horizontal asymptotes of the graph of a rational function algebraically, graphically, and numerically.
- Find the x-intercept(s), y-intercept and coordinates of any hole(s) in the graph of a rational function.
- Graph a rational function by hand and check it by using a graphing utility.
- Solve application problems involving rational functions.
- Exponential and Logarithmic Functions
- Determine if a function is one-to-one and find its inverse if possible.
- Define and graph basic exponential and logarithmic functions (with a strong emphasis on functions with base 10 and e) and their transformations, and identify their domains, ranges, intercepts and asymptotes both by hand and using a graphing utility.
- Solve exponential and logarithmic equations both algebraically and graphically using a graphing utility.
- Use exponential and /or logarithmic functions to solve problems involving compound interest and growth and decay.*
- Expand or condense logarithmic expressions using the properties of logarithms.
- Use the change of base formula to find the logarithm of any base other than 10 or e.
- Use a graphing utility to obtain the power, exponential, logarithmic, or logistic functions of best fit of a given set of values and then determine the model of best fit.
- Solve applied problems involving power, logarithmic, exponential, or logistic regression functions by using a graphing utility.
- Angles and Trigonometric Functions
- Sketch a given angle in standard position.
- Convert angles measured in degrees to radian measure and vice versa.
- Apply the relationship between arc length, area of a sector, central angle, and the radius of a circle and that between linear speed, angular speed, and the radius of a circle.
- State the unit circle definitions and the domain and range of each of the six trigonometric functions for any real number t.
- Find exact values of the six trigonometric functions of special real numbers or of angles in standard position if given partial information.
- Use the Reciprocal Identities, Quotient Identities, Even-Odd Identities, and the Pythagorean Identities to simplify trigonometric expressions.
- Sketch the graphs of the of sine, cosine, and tangent trigonometric functions and their transformations both by hand and by using a graphing utility, stating their periods, amplitudes, and phase shifts if applicable.
- Find an equation for a sinusoidal graph.
- Analytic Trigonometry
- Define the inverse sine, inverse cosine, and inverse tangent functions, and determine their domains and ranges.
- Evaluate the inverse sine, inverse cosine, and inverse tangent function values exactly when possible and, if not possible, approximate using a graphing utility.
- Use properties of inverse functions to find the exact value of certain composite functions and solve equations.
- Determine graphically whether an equation appears to be an identity.
- Use the Quotient Identities, Reciprocal Identities, the Pythagorean Identities, the Even-Odd Identities, the Sum and Difference Identities, the Double Angle Identities, and the Half Angle Identities for sine, cosine and tangent to simplify expressions, to determine exact solutions of equations, or to verify identities.
- Estimate solutions to trigonometric equations by using a graphing utility.
- Applications of Trigonometric Functions
- Evaluate the six trigonometric functions of an acute angle of a right triangle and use these values in applied problems.
- Utilize the Law of Cosines and/or the Law of Sines, including the ambiguous case, to solve an oblique triangle in applied problems.
- Use the formula for the area of a triangle in applied problems.
Assessment and Requirements
- All students will be required to complete a comprehensive final examination that assesses the learning of all course objectives.
- This exam must be weighted in a manner so that this exam score is worth a minimum of fifteen percent (15%) of the final course grade. In selected semesters this exam may be a common exam administered to all sections of Math 175.
- All students will be required to complete online homework. This online homework must be weighted in such a manner so that it is worth between six percent (6%) and twelve percent (12%) of the final course grade.
- Additional assessment of student achievement may include assignments, quizzes, and exams.
- Application problems must not only be included on chapter exams but also on the final exam.
- A graphing calculator is required of each student and the Mathematics Department recommends and uses the TI-84 Plus series.
- Access to an online homework management system is also required.
- Application problems must be covered in all mathematics courses. Every section in any course outline that includes application problems must be covered.
Credit for Prior College-Level Learning
50, CLEP Pre-calculus exam, taken at any certified Testing Center, such as HFC’s Workforce & Professional Development’s Testing Center (313) 317-6600.
A student may receive credit for Math 175 by earning a minimum score of 50 on the CLEP Precalculus exam.