Transcript title

Precalculus I: Functions

Credits

4

Grading mode

Standard letter grades

Total contact hours

40

Lecture hours

40

Prerequisites

MTH 095 or higher (except MTH 098, MTH 102, MTH 105Z, MTH 244, and STAT 243Z) or minimum placement Math Level 18.

Course Description

A course primarily designed for students preparing for trigonometry or calculus. This course focuses on functions and their properties, including polynomial, rational, exponential, logarithmic, piecewise-defined, and inverse functions. These topics will be explored symbolically, numerically, and graphically in real life applications and interpreted in context. This course emphasizes skill building, problem solving, modeling, reasoning, communication, connections with other disciplines, and the appropriate use of present-day technology.

Course learning outcomes

1. Explore the concept of a function numerically, symbolically, verbally, and graphically and identify properties of functions both with and without technology.
2. Analyze polynomial, rational, exponential, and logarithmic functions, as well as piecewise-defined functions, in both algebraic and graphical contexts, and solve equations involving these function types.
3. Demonstrate algebraic and graphical competence in the use and application of functions including notation, evaluation, domain/range, algebraic operations composition, inverses, transformations, symmetry, rate of change, extrema, intercepts, asymptotes, and other behavior.
4. Use variables and functions to represent unknown quantities, create models, find solutions, and communicate an interpretation of the results.
5. Determine the reasonableness and implications of mathematical methods, solutions, and approximations in context.

Content outline

1. Solve Equations Using Graphing Technology
   1. Solve f(t) = g(t) by the intersection method, f(t) = 0 by the root method, and f(t) = k using tables of values.
2. Functions (includes algebraic form, graphic form, tabular form)
   1. Explain the concept of a function. e.g., What is a function? Is a relationship is a function?  Vertical line test
   2. Understand function notation/vocabulary in algebraic, graphic and tabular sense.
   3. Evaluate functions with (a) a change of variable, (b) at a value (c) with a new expression
   4. Give the domain and range of a function from its algebraic, graphic or tabular form.
   5. Give increasing or decreasing intervals
   6. Use appropriate notation to describe an interval
   7. Graph piecewise functions
   8. Rewrite a piecewise graph in algebraic format.
   9. Rewrite an implicit function in explicit form.
   10. Simplify the difference quotient 
   11. Compute the average rate of change
   12. Transform a function graphically
   13. Perform operations with functions
   14. Find the inverse of a function algebraically or graphically
   15. Determine if the inverse of a function is a function. Horizontal line test
   16. Show that f∘f^-1 (x)= f^-1∘f(x)= x algebraically
   17. Distinguish between f^-1 (x) vs. [f(x)]^-1
3. Mathematical Models
   1. Identify the independent and the dependent variable
   2. Use a mathematical model (quadratic, polynomial, rational and exponential) given in an algebraic or graphic form to draw conclusions, make predictions and analyze behavior inherent in the model.
4. Exponential Functions and Exponential Equations (includes algebraic form, graphic form, tabular form)
   1. Analyze an exponential model in algebraic or graphic form.
   2. Analyze P(t) = P₀ e^±kt
   3. Analyze y=a b^t, convert to y=a e^kt 
   4. Solve exponential equations algebraically
5. Logarithms and Logarithmic Equations (includes algebraic form, graphic form, tabular form)
   1. Apply the Rules of Logarithms to simplify expressions
   2. Solve logarithmic equations algebraically.
6. Exponential Applications
   1. Solve problems involving half-life.
   2. Solve problems involving exponential growth: e.g., population growth, time value of money, etc.
7. Modeling Applications
   1. Create mathematical models based on logical reasoning and algebraic relations
   2. Analyze mathematical models, e.g., find the maximum population, find when a population goes extinct, etc.
   3. Choose the appropriate function(s) associated with a graph.
8. Polynomial Functions
   1. Determine the roots and multiplicities for a polynomial in factored form.
   2. Determine if the graph of a polynomial, written in factored form, passes through (slices) or bounces off the x-axis at the roots.
   3. Write a polynomial function in factored form representing a given graph.
   4. Determine the general shape of a polynomial (parabola, cubic, number of turns), written algebraically, from the degree and leading coefficient of the polynomial.
9. Rational Functions
   1. Find asymptotes of rational functions
   2. Find the domain and range of a rational function
   3. Analyzing graphs of rational functions (intercepts, end behavior, and horizontal and vertical asymptotes) with technology and without
10. Writing and Working in a Group
    1. Effectively communicate mathematical concepts in writing using correct mathematical notation
    2. Work collaboratively with their peers on projects or activities to explore mathematical concepts.

Required materials

Students are required to have a license for web-based software which includes an e-text. Paper copy of the textbook is optional.

General education/Related instruction lists

• Science not Lab
• Mathematics