Algebra and Geometry Basics

Basic Concepts:
- Variables: Symbols representing numbers (e.g., x, y).
- Constants: Fixed values (e.g., 2, -5).
- Expressions: Combinations of variables and constants (e.g., 2x + 3).
Equations:
- Linear Equations: Form y = mx + b, where m is the slope and b is the y-intercept.
- Quadratic Equations: Form ax² + bx + c = 0; solved using factoring, completing the square, or quadratic formula.
Functions:
- Definition: A relation where each input has a single output.
- Types: Linear, quadratic, polynomial, exponential, logarithmic.
Factoring:
- Techniques: Grouping, difference of squares, trinomials.
- Importance: Solves equations and simplifies expressions.

Basic Principles:
- Points, Lines, Line segments, and Rays: Fundamental elements of geometry.
- Angles: Measured in degrees; types include acute (<90°), right (90°), obtuse (>90°).
Shapes and Forms:
- Two-dimensional (2D): Circles, triangles, rectangles, quadrilaterals.
  - Area Formulas:
    - Triangle: ( \frac{1}{2} \times base \times height )
    - Rectangle: ( length \times width )
    - Circle: ( \pi \times radius^2 )
- Three-dimensional (3D): Spheres, cubes, cylinders.
  - Volume Formulas:
    - Sphere: ( \frac{4}{3} \pi \times radius^3 )
    - Cube: ( side^3 )
    - Cylinder: ( \pi \times radius^2 \times height )
Theorems:
- Pythagorean Theorem: In right triangles, ( a^2 + b^2 = c^2 ), where c is the hypotenuse.

Descriptive Statistics:
- Measures of Central Tendency: Mean, median, mode.
- Measures of Dispersion: Range, variance, standard deviation.
Probability:
- Definition: Measure of likelihood, ranging from 0 to 1.
- Types: Independent, dependent events; complementary events.
Data Representation:
- Graphs: Bar charts, histograms, pie charts, line graphs.
- Tables: Organize data for easier analysis.
Inferential Statistics:
- Uses sample data to draw conclusions about a population.
- Techniques: Hypothesis testing, confidence intervals, regression analysis.

Variables: Symbols representing unknown numbers.
Constants: Fixed numerical values that don't change.
Expressions: Combinations of variables, constants, and mathematical operations.
Linear Equations: Have the form y = mx + b, where m is the slope and b is the y-intercept.
Quadratic Equations: Have the form ax² + bx + c = 0, solved using factoring, completing the square, or the quadratic formula.

Definition: A relationship where each input has a unique output.
Types: Linear, quadratic, polynomial, exponential, and logarithmic functions.

Grouping: A technique used to factor expressions with four or more terms by grouping terms with common factors together.
Difference of Squares: A technique used to factor expressions in the form a² - b² as (a + b)(a - b).
Trinomials: Expressions with three terms, commonly factored as (ax + b)(cx + d).
Importance of Factoring: Simplifies expressions and helps solve equations.

Basic Elements:
- Points: Locations in space.
- Lines: Extend infinitely in both directions.
- Line Segments: Finite portions of lines.
- Rays: Start at a point and extend infinitely in one direction.
Angles:
- Measured in degrees.
- Acute Angles: Less than 90°.
- Right Angles: Equal to 90°.
- Obtuse Angles: Greater than 90°.

2D Shapes:
- Circles: Closed curves with all points equidistant from a center point.
- Triangles: Three-sided polygons.
- Rectangles: Four-sided polygons with four right angles.
- Quadrilaterals: Four-sided polygons.
Area Formulas:
- Triangle: ( \frac{1}{2} \times base \times height )
- Rectangle: ( length \times width )
- Circle: ( \pi \times radius^2 )
3D Shapes:
- Spheres: Three-dimensional objects with all points equidistant from a center point.
- Cubes: Six-sided shapes with all sides equal in length.
- Cylinders: Three-dimensional objects with two circular bases and parallel sides.
Volume Formulas:
- Sphere: ( \frac{4}{3} \pi \times radius^3 )
- Cube: ( side^3 )
- Cylinder: ( \pi \times radius^2 \times height )

Pythagorean Theorem: In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. ( a^2 + b^2 = c^2 ).

Descriptive Statistics: Summarizes and describes data.
- Measures of Central Tendency:
  - Mean: Average of a dataset.
  - Median: Middle value in a sorted dataset.
  - Mode: Most frequent value in a dataset.
- Measures of Dispersion:
  - Range: Difference between the largest and smallest values in a dataset.
  - Variance: Average squared deviation from the mean.
  - Standard Deviation: Square root of the variance.

Definition: Measures the likelihood of an event occurring between 0 and 1.
- Independent Events: Events that do not influence each other.
- Dependent Events: Events where the outcome of one event affects the outcome of another.
- Complementary Events: Two events that are the only possible outcomes. The probability of one event is 1 minus the probability of the other.

Graphs:
- Bar Charts: Visualize categorical data with rectangular bars.
- Histograms: Visualize numerical data with adjacent bars.
- Pie Charts: Visualize proportions of a whole.
- Line Graphs: Show trends in data over time.
Tables: Organize data for easier analysis and interpretation.

Uses: Draws conclusions about a population based on a sample.
Techniques:
- Hypothesis Testing: Uses sample data to test a claim about a population.
- Confidence Intervals: Estimate a range of values for a population parameter.
- Regression Analysis: Examines the relationship between variables.