College Algebra: Functions, Equations, and Applications – Study Guide

Course Overview

This study guide covers the main topics outlined in a College Algebra course, focusing on functions (including rational, polynomial, exponential, and logarithmic), equations, and systems of linear equations. The guide is organized by major topics and subtopics as presented in the syllabus.

Complex Numbers

Definition and Properties

Complex numbers extend the real number system and are essential for solving equations that do not have real solutions.

• Complex Number: A number of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit ($i^2 = -1$).

• Operations: Addition, subtraction, multiplication, and division follow specific rules for combining real and imaginary parts.

• Example: $(2 + 3i) + (1 - 4i) = 3 - i$

Quadratic Equations

Solving Quadratic Equations

Quadratic equations are second-degree polynomial equations and can be solved using various methods.

• Standard Form: $ax^2 + bx + c = 0$

• Factoring: Expressing the quadratic as a product of two binomials.

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

• Completing the Square: Rewriting the equation to isolate $x$.

• Example: Solve $x^2 - 5x + 6 = 0$ by factoring: $(x-2)(x-3)=0$ so $x=2$ or $x=3$.

Other Types of Equations and Applications

Absolute Value Equations and Inequalities

Equations involving absolute values require considering both positive and negative cases.

• Definition: $|x| = a$ implies $x = a$ or $x = -a$

• Example: $|x - 3| = 5$ yields $x = 8$ or $x = -2$

Rectangular Coordinates and Graphs

Cartesian Plane and Graphing

The rectangular coordinate system is used to graph equations and visualize relationships between variables.

• Axes: The $x$-axis (horizontal) and $y$-axis (vertical) intersect at the origin $(0,0)$.

• Plotting Points: Each point is represented as $(x, y)$.

• Example: The point $(3, -2)$ is 3 units right and 2 units down from the origin.

Functions

Definition and Types

Functions describe relationships between input and output values, and are central to algebra.

• Function: A rule that assigns each input exactly one output.

• Notation: $f(x)$ denotes the output when $x$ is the input.

• Types: Linear, quadratic, polynomial, rational, exponential, logarithmic.

• Example: $f(x) = 2x + 1$ is a linear function.

Linear Functions and Models

Equations of Lines

Linear functions are represented by straight lines and are used to model constant rates of change.

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Example: A line with slope 3 passing through $(1,2)$: $y - 2 = 3(x - 1)$

Graphs of Basic Functions

Common Function Graphs

Understanding the shapes of basic function graphs aids in analysis and problem-solving.

• Linear: Straight line

• Quadratic: Parabola opening up or down

• Exponential: Rapid increase or decrease

• Logarithmic: Slow increase, passes through $(1,0)$

Function Operations and Composition

Combining Functions

Functions can be added, subtracted, multiplied, divided, or composed to create new functions.

• Sum: $(f + g)(x) = f(x) + g(x)$

• Composition: $(f \circ g)(x) = f(g(x))$

• Example: If $f(x) = x^2$ and $g(x) = x + 1$, then $(f \circ g)(x) = (x + 1)^2$

Quadratic Functions and Models

Vertex and Axis of Symmetry

Quadratic functions have a characteristic U-shaped graph and are used to model various phenomena.

• Vertex: The highest or lowest point, given by $x = -\frac{b}{2a}$

• Axis of Symmetry: Vertical line through the vertex

• Example: For $f(x) = x^2 - 4x + 3$, vertex at $x = 2$

Polynomial Functions

Zeros and Synthetic Division

Polynomial functions generalize quadratics and can be analyzed for their roots and behavior.

• Zero: A value $x$ where $f(x) = 0$

• Synthetic Division: A shortcut for dividing polynomials by linear factors

• Example: Divide $x^3 - 6x^2 + 11x - 6$ by $x - 1$ using synthetic division

Rational Functions

Definition and Properties

Rational functions are quotients of polynomials and have unique features such as asymptotes.

• Form: $f(x) = \frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$

• Vertical Asymptotes: Values where $Q(x) = 0$

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

Exponential and Logarithmic Functions

Definitions and Applications

Exponential and logarithmic functions model growth, decay, and many real-world phenomena.

• Exponential Function: $f(x) = a^x$ where $a > 0$ and $a \neq 1$

• Logarithmic Function: $f(x) = \log_a(x)$, the inverse of the exponential function

• Properties: $\log_a(a^x) = x$ and $a^{\log_a(x)} = x$

• Example: $f(x) = 2^x$ and $g(x) = \log_2(x)$

Evaluating Logarithms and the Change of Base Formula

Calculating Logarithms

Logarithms can be evaluated using properties and the change of base formula.

• Change of Base Formula: $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$

• Example: $\log_2(8) = \frac{\log_{10}(8)}{\log_{10}(2)}$

Exponential and Logarithmic Equations

Solving Equations

Equations involving exponentials and logarithms are solved using properties and algebraic manipulation.

• Exponential Equation: $a^{f(x)} = a^{g(x)}$ implies $f(x) = g(x)$

• Logarithmic Equation: $\log_a(f(x)) = \log_a(g(x))$ implies $f(x) = g(x)$

• Example: Solve $2^x = 8$; $x = 3$

Matrix Solution of Linear Systems

Solving Systems Using Matrices

Matrices provide a systematic way to solve systems of linear equations.

• Matrix Form: $AX = B$ where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix

• Determinant: Used to determine if a unique solution exists

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

Summary Table: Main Function Types

Function Type	General Form	Graph Shape	Key Properties
Linear	$y = mx + b$	Straight line	Constant rate of change
Quadratic	$y = ax^2 + bx + c$	Parabola	Vertex, axis of symmetry
Polynomial	$y = a_nx^n + \ldots + a_0$	Varied	Degree, zeros
Rational	$y = \frac{P(x)}{Q(x)}$	Hyperbola, asymptotes	Vertical/horizontal asymptotes
Exponential	$y = a^x$	Rapid increase/decrease	Growth/decay
Logarithmic	$y = \log_a(x)$	Slow increase	Inverse of exponential

Additional info: Some subtopics (e.g., synthetic division, matrix solution) were expanded for completeness and academic context.