Math 0031: College Algebra - Schedule and Practice Problems Overview

Course Overview

This document outlines the schedule and recommended practice problems for a College Algebra course. The topics align with standard College Algebra curriculum, covering functions, equations, inequalities, polynomials, rational functions, exponential and logarithmic functions, systems of equations, conic sections, and sequences and series.

Course Topics and Structure

Graphs, Functions, and Models

• Definition of a Function: A function is a relation in which each input (domain) has exactly one output (range).

• Graphing Functions: Understanding how to plot basic functions and interpret their graphs.

• Modeling with Functions: Using functions to represent real-world situations.

• Example: The function $f(x) = 2x + 3$ is a linear function whose graph is a straight line.

More on Functions

• Domain and Range: Identifying the set of possible inputs and outputs for a function.

• Function Operations: Addition, subtraction, multiplication, division, and composition of functions.

• Inverse Functions: A function that 'undoes' the action of the original function.

• Example: If $f(x) = 2x + 1$, then its inverse is $f^{-1}(x) = \frac{x - 1}{2}$.

Quadratic Functions and Equations; Inequalities

• Quadratic Functions: Functions of the form $f(x) = ax^2 + bx + c$.

• Solving Quadratic Equations: Methods include factoring, completing the square, and the quadratic formula.

• Quadratic Formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Inequalities: Solving and graphing linear and quadratic inequalities.

• Example: Solve $x^2 - 5x + 6 < 0$ by factoring and testing intervals.

Polynomial Functions and Rational Functions

• Polynomial Functions: Functions involving terms with non-negative integer exponents.

• End Behavior: Determined by the leading term of the polynomial.

• Rational Functions: Functions of the form $f(x) = \frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$.

• Asymptotes: Vertical and horizontal asymptotes describe the behavior of rational functions at extreme values.

• Example: $f(x) = \frac{1}{x-2}$ has a vertical asymptote at $x=2$.

Exponential Functions and Logarithmic Functions

• Exponential Functions: Functions of the form $f(x) = a \cdot b^x$.

• Logarithmic Functions: The inverse of exponential functions, $f(x) = \log_b x$.

• Properties of Logarithms: Product, quotient, and power rules.

• Example: $\log_b (xy) = \log_b x + \log_b y$.

Systems of Equations and Matrices

• Solving Systems: Methods include substitution, elimination, and using matrices.

• Matrices: Rectangular arrays of numbers used to represent and solve systems.

• Example: Solve the system $\begin{cases} x + y = 3 \\ 2x - y = 0 \end{cases}$ using elimination.

Conic Sections

• Definition: Curves obtained by intersecting a plane with a double-napped cone: circles, ellipses, parabolas, and hyperbolas.

• Standard Equations:

Circle: $(x-h)^2 + (y-k)^2 = r^2$ Ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ Parabola: $y = a(x-h)^2 + k$ Hyperbola: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

Sequences, Series, and Combinatorics

• Sequences: Ordered lists of numbers, such as arithmetic or geometric sequences.

• Series: The sum of the terms of a sequence.

• Arithmetic Sequence Formula:

$a_n = a_1 + (n-1)d$

• Geometric Sequence Formula:

$a_n = a_1 r^{n-1}$

• Combinatorics: Counting principles, permutations, and combinations.

• Example: The number of ways to choose $r$ objects from $n$ is $\binom{n}{r} = \frac{n!}{r!(n-r)!}$.

Practice Problems and Schedule

The document lists specific textbook sections and recommended problems for each topic. Students are encouraged to follow the schedule and complete the practice problems to reinforce their understanding of each concept.

Additional info: This summary is based on the schedule and practice problems list, which outlines the main topics and recommended exercises for a College Algebra course. The above content expands on the listed topics to provide a self-contained study guide.