Algebra and Geometry Basics

Algebra

• 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.

Geometry

• 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.

Statistics

• 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.

Algebra

• 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.

Functions

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

Factoring

• 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.

Geometry

• 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°.

Shapes and Forms

• 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 )

Theorems

• 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 ).

Statistics

• 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.

Probability

• 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.

Data Representation

• 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.

Inferential Statistics

• 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.