📘 Ultimate College Math Study Guide
Table of Contents
1. Algebra
2. Functions and Graphs
3. Trigonometry
4. Pre-Calculus
5. Calculus
6. Key Formulas & Cheat Sheet
7. Practice Questions & Answers
8. Study Tips & Strategies
1. Algebra
1.1 Basic Concepts
● Variables and constants: Symbols representing numbers.
● Expressions: Combination of variables, numbers, and operations.
● Equations: Statements that two expressions are equal.
Example: Solve for x:
2x + 5 = 13
Solution:
2x = 13 − 5 → 2x = 8 → x = 4
1.2 Factoring
● Factoring quadratics: a² + 5a + 6 = (a + 2)(a + 3)
● Difference of squares: a² − b² = (a − b)(a + b)
● Factoring by grouping: ab + ac + bd + bc = (a + b)(c + d)
Practice Problem: Factor x² − 9
, 1.3 Solving Equations
● Linear equations: ax + b = c
● Quadratic equations: ax² + bx + c = 0 → use factoring or quadratic formula
Quadratic formula:
x = [−b ± √(b² − 4ac)] / 2a
Practice Problem: Solve x² − 5x + 6 = 0
2. Functions and Graphs
2.1 Functions
● Definition: A relation where each input has exactly one output.
● Notation: f(x), g(x), h(x)
Example: f(x) = 2x + 3
● f(2) = 2(2) + 3 = 7
2.2 Graphs
● Linear function: y = mx + b → slope = m, y-intercept = b
● Quadratic function: y = ax² + bx + c → parabola shape
● Exponential function: y = a·b^x → growth/decay
Practice Problem: Graph y = x² − 4x + 3
3. Trigonometry
3.1 Basic Ratios
● sin(θ) = opposite / hypotenuse
● cos(θ) = adjacent / hypotenuse
● tan(θ) = opposite / adjacent
Practice Problem: Find sin(30°), cos(60°), tan(45°)
Table of Contents
1. Algebra
2. Functions and Graphs
3. Trigonometry
4. Pre-Calculus
5. Calculus
6. Key Formulas & Cheat Sheet
7. Practice Questions & Answers
8. Study Tips & Strategies
1. Algebra
1.1 Basic Concepts
● Variables and constants: Symbols representing numbers.
● Expressions: Combination of variables, numbers, and operations.
● Equations: Statements that two expressions are equal.
Example: Solve for x:
2x + 5 = 13
Solution:
2x = 13 − 5 → 2x = 8 → x = 4
1.2 Factoring
● Factoring quadratics: a² + 5a + 6 = (a + 2)(a + 3)
● Difference of squares: a² − b² = (a − b)(a + b)
● Factoring by grouping: ab + ac + bd + bc = (a + b)(c + d)
Practice Problem: Factor x² − 9
, 1.3 Solving Equations
● Linear equations: ax + b = c
● Quadratic equations: ax² + bx + c = 0 → use factoring or quadratic formula
Quadratic formula:
x = [−b ± √(b² − 4ac)] / 2a
Practice Problem: Solve x² − 5x + 6 = 0
2. Functions and Graphs
2.1 Functions
● Definition: A relation where each input has exactly one output.
● Notation: f(x), g(x), h(x)
Example: f(x) = 2x + 3
● f(2) = 2(2) + 3 = 7
2.2 Graphs
● Linear function: y = mx + b → slope = m, y-intercept = b
● Quadratic function: y = ax² + bx + c → parabola shape
● Exponential function: y = a·b^x → growth/decay
Practice Problem: Graph y = x² − 4x + 3
3. Trigonometry
3.1 Basic Ratios
● sin(θ) = opposite / hypotenuse
● cos(θ) = adjacent / hypotenuse
● tan(θ) = opposite / adjacent
Practice Problem: Find sin(30°), cos(60°), tan(45°)
