Key Concepts in Mathematics

Arithmetic: Basic operations (addition, subtraction, multiplication, division).
Algebra: Study of symbols and the rules for manipulating those symbols; includes solving equations.
Geometry: Study of shapes, sizes, and properties of space; involves points, lines, angles, surfaces, and solids.
Trigonometry: Deals with the relationships between the angles and sides of triangles; includes sine, cosine, and tangent functions.
Calculus: Study of change and motion; involves differentiation (rates of change) and integration (accumulating quantities).
Statistics: Collection, analysis, interpretation, presentation, and organization of data.
Probability: Study of uncertainty and the likelihood of events occurring.

Pythagorean Theorem: In a right triangle, (a^2 + b^2 = c^2), where (c) is the hypotenuse.
Fundamental Theorem of Algebra: Every non-constant polynomial has at least one complex root.
Calculus:
- Mean Value Theorem: A function that is continuous on a closed interval and differentiable on the open interval has at least one point where the derivative equals the average rate of change.
- Fundamental Theorem of Calculus: Relates differentiation and integration, stating that the integral of a function can be computed using its antiderivative.

Order of Operations: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right) (PEMDAS/BODMAS).
Properties of Numbers:
- Commutative Property: (a + b = b + a) and (ab = ba).
- Associative Property: ((a + b) + c = a + (b + c)) and ((ab)c = a(bc)).
- Distributive Property: (a(b + c) = ab + ac).

Function: A relation that assigns exactly one output for each input.
Types of Functions:
- Linear: (f(x) = mx + b) (graph is a straight line).
- Quadratic: (f(x) = ax^2 + bx + c) (graph is a parabola).
- Exponential: (f(x) = a \cdot b^x) (growth or decay).
Graphing: Understanding the Cartesian plane, plotting points, and analyzing slope and intercepts.

Measures of Central Tendency:
- Mean: Average of a set of numbers.
- Median: Middle value when numbers are sorted.
- Mode: Most frequently occurring value.
Variance and Standard Deviation: Measure of spread in a data set.

Experiments and Outcomes: An experiment is a process with uncertain results; outcomes are the possible results.
Events: A set of outcomes; can be mutually exclusive or independent.
Probability Formula: (P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}).

Arithmetic: Involves the basic operations such as addition, subtraction, multiplication, and division.
Algebra: Focuses on symbols and the rules for manipulating them, including solving equations.
Geometry: Examines shapes, sizes, and properties of space, covering elements like points, lines, angles, surfaces, and solids.
Trigonometry: Explores relationships between the angles and sides of triangles using functions like sine, cosine, and tangent.
Calculus: Investigates change and motion through differentiation (rates of change) and integration (accumulating quantities).
Statistics: Involves collecting, analyzing, interpreting, presenting, and organizing data.
Probability: Studies uncertainty and the likelihood of occurrence of various events.

Pythagorean Theorem: In a right triangle, the relationship is defined by (a^2 + b^2 = c^2), where (c) is the hypotenuse.
Fundamental Theorem of Algebra: Asserts that every non-constant polynomial equation has at least one complex root.
Mean Value Theorem: States that for a continuous and differentiable function on a closed interval, there exists at least one point where the derivative equals the average rate of change.
Fundamental Theorem of Calculus: Connects differentiation and integration, indicating that the integral can be calculated using the function's antiderivative.

Order of Operations: Follows the sequence of Parentheses, Exponents, Multiplication, Division (left to right), Addition, and Subtraction (PEMDAS/BODMAS).
Commutative Property: Shows that addition and multiplication are order-independent, expressed as (a + b = b + a) and (ab = ba).
Associative Property: Indicates that grouping does not affect the result for addition and multiplication, expressed as ((a + b) + c = a + (b + c)).
Distributive Property: Demonstrates how to distribute multiplication over addition, expressed as (a(b + c) = ab + ac).

Function: Defined as a relation that assigns exactly one output for each input.
Types of Functions:
- Linear: Represents a straight line with the form (f(x) = mx + b).
- Quadratic: Takes the form (f(x) = ax^2 + bx + c) and graphs as a parabola.
- Exponential: Expressed as (f(x) = a \cdot b^x) representing growth or decay.
Graphing: Involves understanding the Cartesian plane, plotting points, and analyzing key features like slope and intercepts.

Mean: The average value calculated by summing all numbers and dividing by the count.
Median: The middle value in a sorted list of numbers, representing the central tendency.
Mode: The most frequently occurring value within a data set.
Variance and Standard Deviation: Measures that indicate the spread and variability within a data set.

Experiments and Outcomes: An experiment yields uncertain results, with outcomes being the possible results derived from it.
Events: Comprise sets of outcomes that can either be mutually exclusive or independent.
Probability Formula: Expressed as (P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}), quantifying the likelihood of an event.

Engage in regular practice to boost problem-solving capabilities.
Prioritize understanding concepts rather than rote memorization of procedures.
Utilize visual aids, such as graphs and diagrams, to clarify complex ideas.
Decompose difficult problems into smaller, manageable components for easier resolution.