College Algebra Final Exam Review: Comprehensive Study Notes

Equations & Inequalities

Solving Equations Over the Complex Numbers

Solving equations is a foundational skill in algebra. Equations may be linear, quadratic, or involve higher powers, and solutions may be real or complex numbers.

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

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

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

• Complex Solutions: If $b^2 - 4ac < 0$, solutions are complex numbers.

• Example: Solve $x^2 + 1 = 0$. Solution: $x = i$ or $x = -i$.

Solving Inequalities

Inequalities involve finding the set of values that satisfy a given relation, such as $ax + b > 0$ or $|x - a| < b$.

• Interval Notation: Solutions are often expressed in interval notation, e.g., $(a, b)$ or $[a, b]$.

• Absolute Value Inequalities: Solve $|x - a| < b$ by considering $-b < x - a < b$.

• Example: Solve $2x - 3 < 5$. Solution: $x < 4$.

Graphs of Equations & Functions

Graphing Basic and Transformed Functions

Understanding how to graph functions and their transformations is essential for visualizing algebraic relationships.

• Parent Function: The simplest form of a function, e.g., $f(x) = x^2$.

• Transformations: Include shifts, stretches, compressions, and reflections.

• Example: $f(x) = |x|$ is the parent absolute value function. $f(x) = |x - 2| + 3$ is shifted right by 2 and up by 3.

Finding Intercepts, Vertex, and Axis of Symmetry

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

• Vertex of a Parabola: For $y = ax^2 + bx + c$, vertex at $x = -\frac{b}{2a}$.

• Axis of Symmetry: Vertical line $x = -\frac{b}{2a}$.

Functions

Definition and Properties

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

• Domain: Set of all possible input values.

• Range: Set of all possible output values.

• Inverse Function: If $f(x)$ is one-to-one, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$.

Operations with Functions

• Sum, Difference, Product, Quotient: $(f+g)(x) = f(x) + g(x)$, $(f-g)(x) = f(x) - g(x)$, $(fg)(x) = f(x)g(x)$, $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ (where $g(x) \neq 0$).

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

Polynomial and Rational Functions

Factoring and Roots

• Factoring: Expressing a polynomial as a product of its factors.

• Roots/Zeros: Values of $x$ for which $f(x) = 0$.

• Example: Factor $f(x) = 2x^3 - 3x^2 - 5x + 6$ into linear factors.

Asymptotes and End Behavior

• Vertical Asymptotes: Values of $x$ where the function is undefined (denominator zero).

• Horizontal Asymptotes: Describe the behavior as $x \to \infty$ or $x \to -\infty$.

• Example: For $f(x) = \frac{2x}{x^2 - 1}$, vertical asymptotes at $x = 1, -1$.

Exponential & Logarithmic Functions

Properties and Applications

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

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

• Properties: $\log_a (xy) = \log_a x + \log_a y$, $\log_a (\frac{x}{y}) = \log_a x - \log_a y$, $\log_a (x^k) = k \log_a x$.

• Applications: Population growth, compound interest, and radioactive decay.

• Example: The population $P(t) = P_0 e^{kt}$ models exponential growth.

Systems of Equations & Matrices

Solving Systems of Equations

• Substitution and Elimination: Methods for solving systems of linear equations.

• Matrix Method: Systems can be written as $AX = B$ and solved using matrix operations.

• Determinant: For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, determinant is $ad - bc$.

Applications

• Mixture Problems: Setting up equations to solve for quantities in mixtures.

• Distance Problems: Using the formula $\text{Distance} = \text{Rate} \times \text{Time}$.

Conic Sections

Circles

• Standard Form: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

• Finding Center and Radius: Complete the square if necessary to rewrite in standard form.

Sequences, Series, & Induction

Arithmetic and Geometric Sequences

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• Summation Notation: $\sum_{k=1}^n a_k$

• Example: Find the first five terms of $a_n = 2^n$.

Series

• Sum of Arithmetic Series: $S_n = \frac{n}{2}(a_1 + a_n)$

• Sum of Geometric Series: $S_n = a_1 \frac{1 - r^n}{1 - r}$, $r \neq 1$

Combinatorics & Probability

Basic Counting Principles

• Permutations: Number of ways to arrange $n$ objects: $n!$

• Combinations: Number of ways to choose $r$ objects from $n$: $C(n, r) = \frac{n!}{r!(n-r)!}$

Summary Table: Key Algebraic Concepts

Topic	Key Formula/Concept	Example
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	$x^2 + 2x + 1 = 0$
Exponential Growth	$P(t) = P_0 e^{kt}$	Population doubling
Circle Equation	$(x - h)^2 + (y - k)^2 = r^2$	Center (2, 3), $r = 5$
Arithmetic Sequence	$a_n = a_1 + (n-1)d$	$a_1 = 2, d = 3$
Matrix Determinant	$ad - bc$	$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

Additional info:

• These notes are based on a comprehensive final exam review covering all major College Algebra topics, including equations, inequalities, functions, graphing, conic sections, systems of equations, matrices, sequences, series, and applications.

• Students should practice solving each type of problem and understand the underlying concepts and formulas.