College Algebra Comprehensive Study Guide

Equations and Inequalities

Solving Linear and Quadratic Equations

Solving equations is a foundational skill in algebra, involving finding the value(s) of the variable that make the equation true.

• Linear Equations: Equations of the form $ax + b = 0$.

• Quadratic Equations: Equations of the form $ax^2 + bx + c = 0$.

• Factoring, quadratic formula, and completing the square are common methods for solving quadratics.

• Example: Solve $2x^2 - 3x - 2 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Solving Rational and Radical Equations

• Rational Equations: Equations involving fractions with polynomials in the numerator and denominator.

• Radical Equations: Equations involving roots, such as square roots.

• Example: Solve $\frac{36}{x-2} + \frac{2x}{x+2} = 8$ by finding a common denominator.

Functions and Graphs

Definition and Properties of Functions

A function is a relation in which each input has exactly one output. Functions can be represented by equations, tables, or graphs.

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Even/Odd Functions: Even if $f(-x) = f(x)$, odd if $f(-x) = -f(x)$.

• Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Graphing Functions

• Plot points by substituting values for $x$ and finding corresponding $y$ values.

• Identify intercepts, asymptotes, and general shape.

• Example: Graph $f(x) = x^2 - 2$ by plotting several points and drawing the parabola.

Transformations of Functions

• Vertical and horizontal shifts, reflections, and stretches/compressions.

• Example: $f(x) = (x-2)^2 + 1$ is $f(x) = x^2$ shifted right 2 units and up 1 unit.

Polynomial and Rational Functions

Polynomial Functions

• Functions of the form $f(x) = a_nx^n + \ldots + a_1x + a_0$.

• Degree determines the end behavior and number of possible real roots.

• Example: $f(x) = x^3 - 4x$ is a cubic polynomial.

Rational Functions

• Functions of the form $f(x) = \frac{p(x)}{q(x)}$ where $q(x) \neq 0$.

• Vertical asymptotes occur where $q(x) = 0$; horizontal asymptotes depend on the degrees of $p(x)$ and $q(x)$.

• Example: $f(x) = \frac{2x-3}{x^2-5x+6}$ has vertical asymptotes at $x=2$ and $x=3$.

Exponential and Logarithmic Functions

Exponential Functions

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

• Used to model growth and decay, such as population or compound interest.

• Example: $P(t) = 2500e^{0.07t}$ models population growth.

Logarithmic Functions

• Inverse of exponential functions: $y = \log_b x$ means $b^y = x$.

• Properties: $\log_b(xy) = \log_b x + \log_b y$, $\log_b(x/y) = \log_b x - \log_b y$, $\log_b(x^k) = k\log_b x$.

• Example: Solve $\log_2 x = 5$ for $x$.

Systems of Equations and Inequalities

Solving Systems of Linear Equations

• Methods: Substitution, elimination, and graphing.

• Systems can have one solution, no solution, or infinitely many solutions.

• Example: Solve $\begin{cases} x + y = 2 \\ x - y = 4 \end{cases}$ by elimination.

Applications of Systems

• Word problems involving mixtures, investments, and optimization.

• Example: A hotel rents rooms with and without kitchens at different rates; set up a system to find the number of each type rented.

Matrices and Determinants

Matrix Operations

• Addition, subtraction, and multiplication of matrices.

• Determinants are used to solve systems of equations and find inverses.

• Example: For $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $\det(A) = 1 \cdot 4 - 2 \cdot 3 = -2$.

Conic Sections

Parabolas, Circles, Ellipses, and Hyperbolas

• Standard equations for each conic section.

• Identify and graph conic sections from their equations.

• Example: The equation $y^2 = 4px$ represents a parabola.

Sequences, Induction, and Probability

Arithmetic and Geometric Sequences

• Arithmetic: $a_n = a_1 + (n-1)d$

• Geometric: $a_n = a_1 r^{n-1}$

• Example: Find the 10th term of the sequence $2, 4, 8, 16, \ldots$

Probability Basics

• Probability of an event: $P(E) = \frac{\text{number of favorable outcomes}}{\text{total outcomes}}$

• Applications in counting and arrangements.

Additional Topics

Compound Interest

• Formula: $A = P\left(1 + \frac{r}{n}\right)^{nt}$

• Used to calculate future value of investments.

• Example: Compare yields for different compounding periods.

Inverse Functions

• Find $f^{-1}(x)$ by solving $y = f(x)$ for $x$ and interchanging $x$ and $y$.

• Example: If $f(x) = \frac{x+3}{5x-2}$, solve for $f^{-1}(x)$.

Sample Table: Properties of Logarithms

Property	Equation
Product Rule	$\log_b(xy) = \log_b x + \log_b y$
Quotient Rule	$\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$
Power Rule	$\log_b(x^k) = k\log_b x$
Change of Base	$\log_b x = \frac{\log_a x}{\log_a b}$

Additional info: This guide covers all major topics from equations and inequalities to functions, systems, matrices, conic sections, and sequences, as reflected in the provided exam study guide.