Comprehensive Study Notes for College Algebra: Equations, Functions, Graphs, and More

Solving Equations

Types of Equations

Solving equations is a foundational skill in College Algebra. Equations can be classified by their degree and structure, and each type requires specific methods for solution.

• Linear Equations: Equations of the form $ax + b = 0$. Solution: $x = -\frac{b}{a}$.

• Quadratic Equations: Equations of the form $ax^2 + bx + c = 0$. Solution via factoring, completing the square, or the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

• Absolute Value Equations: Equations involving $|x|$. Solve by considering both $x$ and $-x$ cases.

• Absolute Value Inequalities: Inequalities involving $|x|$. Split into two cases and solve each.

• Square Root / Cube Root Equations: Equations involving radicals. Isolate the radical and raise both sides to the appropriate power.

• Logarithmic Equations: Equations involving logarithms. Use properties of logarithms to combine and solve.

• Exponential Equations: Equations involving exponents. Use logarithms to solve for the variable.

Word Problems

Word problems require translating real-world scenarios into algebraic equations, then solving for the unknowns.

• Mixture Problems: Involve combining solutions of different concentrations.

• Prediction Problems: Use algebraic models to predict future values.

Graphing

Types of Functions and Their Graphs

Graphing is essential for visualizing equations and understanding their behavior. Different types of functions have characteristic graphs.

• Parent Functions: Basic functions such as $y = x$, $y = x^2$, $y = |x|$, $y = \sqrt{x}$, $y = \frac{1}{x}$, $y = e^x$, $y = \log x$.

• Linear Functions: Straight lines, $y = mx + b$.

• Quadratic Functions: Parabolas, $y = ax^2 + bx + c$.

• Absolute Value Functions: V-shaped graphs, $y = |x|$.

• Exponential and Logarithmic Functions: Curves showing rapid growth or decay, or slow increase.

• Rational Functions: Functions of the form $y = \frac{P(x)}{Q(x)}$.

• Polynomial Functions: Graphs depend on degree and leading coefficient.

Graph Features

• Intercepts: Points where the graph crosses the axes.

• Vertex: The highest or lowest point of a parabola.

• Asymptotes: Lines the graph approaches but never touches (vertical, horizontal, or slant).

• End Behavior: Describes how the graph behaves as $x \to \infty$ or $x \to -\infty$.

Function Variations

• Direct Variation: $y = kx$

• Inverse Variation: $y = \frac{k}{x}$

• Joint Variation: $y = kxz$

• Combined Variation: Involves more than one type of variation.

Functions

Composition of Functions

Function composition involves applying one function to the result of another.

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

• Example: If $f(x) = 2x$ and $g(x) = x + 3$, then $(f \circ g)(x) = f(g(x)) = 2(x + 3) = 2x + 6$.

Difference Quotient

The difference quotient is used to measure the average rate of change of a function.

• Formula: $\frac{f(x + h) - f(x)}{h}$

• Application: Fundamental in calculus for defining the derivative.

Linear Functions and Slope

Slope and Intercept

The slope and intercept describe the steepness and position of a line.

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Y-intercept: The point where the line crosses the y-axis, $b$ in $y = mx + b$.

• Standard Form: $Ax + By = C$

• Slope-Intercept Form: $y = mx + b$

Systems of Equations

Solving Systems

Systems of equations can be solved using various methods, including substitution, elimination, and matrix techniques.

• Cramer's Rule: Uses determinants to solve systems of linear equations.

• Types: Linear and non-linear systems.

Matrices

Matrix Operations

Matrices are used to organize and solve systems of equations.

• Determinant: A scalar value that can be computed from a square matrix and is used in solving systems.

• Matrix Arithmetic: Addition, subtraction, and multiplication of matrices.

Polynomials

Synthetic Division and Roots

Synthetic division is a shortcut method for dividing polynomials, especially useful for finding roots.

• Synthetic Division: Used to divide a polynomial by a binomial of the form $(x - c)$.

• Finding Roots: Set the polynomial equal to zero and solve for $x$.

Inverse Functions

Finding an Inverse Function

The inverse of a function reverses the roles of input and output.

• Method: Replace $f(x)$ with $y$, swap $x$ and $y$, and solve for $y$.

• Example: If $f(x) = 2x + 3$, then $y = 2x + 3$, swap to $x = 2y + 3$, solve for $y$: $y = \frac{x - 3}{2}$.

Sequences and Series

Introduction to Sequences and Series

Sequences are ordered lists of numbers, and series are sums of sequences.

• Arithmetic Sequence: Each term differs from the previous by a constant, $a_n = a_1 + (n-1)d$.

• Geometric Sequence: Each term is multiplied by a constant ratio, $a_n = a_1 r^{n-1}$.

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