Algebra: Equations and Geometric Properties

• Algebra involves symbols and rules for manipulating those symbols
• It's a broad part of mathematics, useful for solving equations and expressing relationships

Linear Equations

• Linear equations can be written in the form ax + b = 0, where x is a variable and a and b are constants
• Solving linear equations involves finding the value of x that satisfies the equation, using operations that maintain equality
• Graphically, linear equations in two variables represent a straight line on a coordinate plane
• The slope-intercept form of a linear equation is y = mx + b, where m denotes the slope and b is the y-intercept

Shapes and Properties

• Geometry deals with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs
• Basic shapes include points, lines, angles, triangles, quadrilaterals, circles, and polygons
• Each shape has unique properties, such as the sum of angles in a triangle being 180 degrees
• Area and volume calculations are a key part of geometry

Triangles

• Triangles are classified based on their sides (equilateral, isosceles, scalene) and angles (acute, obtuse, right)
• The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides
• Trigonometry focuses on the relationships between the angles and sides of triangles

Trigonometric Functions

• Trigonometric functions (sine, cosine, tangent) relate angles of a right triangle to ratios of its sides
• Sine (sin) is opposite over hypotenuse, cosine (cos) is adjacent over hypotenuse, and tangent (tan) is opposite over adjacent
• These functions can be extended to any angle using the unit circle
• Inverse trigonometric functions (arcsin, arccos, arctan) find the angle given a trigonometric ratio
• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables

Data Analysis

• Statistics involves collecting, analyzing, interpreting, presenting, and organizing data.
• Descriptive statistics summarize and describe the main features of a dataset
• Measures of central tendency include mean (average), median (middle value), and mode (most frequent value)
• Measures of dispersion include range (difference between max and min), variance, and standard deviation (measures the spread of data around the mean)
• Data can be visualized using histograms, box plots, scatter plots, and other graphical representations

Probability

• Probability quantifies the likelihood of an event occurring
• It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty
• Basic probability calculations involve dividing the number of favorable outcomes by the total number of possible outcomes

Random variables

• A random variable is a variable whose value is a numerical outcome of a random phenomenon
• Random variables can be discrete (taking on a countable number of values) or continuous (taking on any value within a range)
• Probability distributions describe the likelihood of each possible value of a random variable
• The expected value of a random variable is the weighted average of its possible values, where the weights are the probabilities of those values