Algebra: Equations, Inequalities, and Functions

Algebra

Equations and Inequalities

• An equation is a statement that says two mathematical expressions are equal
• A solution to an equation is a value that makes the equation true
• Linear equations: ax + by = c, where a, b, and c are constants
• Quadratic equations: ax^2 + bx + c = 0, where a, b, and c are constants
• Inequalities: statements that compare two expressions using >, <, ≥, or ≤

Functions

• A function is a relation between a set of inputs (domain) and a set of possible outputs (range)
• Functions can be represented algebraically, graphically, or numerically
• Domain and range of a function can be restricted or extended
• Types of functions: linear, quadratic, polynomial, exponential, trigonometric

Graphing

• Graphing is a visual representation of a function or relation
• Coordinates (x, y) are used to plot points on a graph
• Graphs can be used to identify key features: intercepts, asymptotes, maxima/minima
• Graphing methods: plotting points, using tables, and graphing calculators

Geometry

Points, Lines, and Planes

• A point is a location in space, represented by a set of coordinates
• A line is a set of points extending infinitely in two directions
• A plane is a flat surface that extends infinitely in all directions
• Intersections: points, lines, and planes can intersect at single points or along lines

Angles and Measurements

• Angles: formed by two rays sharing a common endpoint (vertex)
• Angle measurements: degrees, radians, and right angles (90°)
• Types of angles: acute, obtuse, straight, and reflex
• Properties of angles: supplementary, complementary, and corresponding

Shapes and Solids

• Two-dimensional shapes: triangles, quadrilaterals, polygons, circles
• Three-dimensional shapes: pyramids, prisms, spheres, cylinders
• Properties of shapes: congruence, similarity, and symmetry
• Formulas for perimeter, area, and volume of various shapes

Algebra

Equations

• An equation is a statement that says two mathematical expressions are equal.
• A solution to an equation is a value that makes the equation true.

Linear Equations

• The general form of a linear equation is ax + by = c, where a, b, and c are constants.

Quadratic Equations

• The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.

Inequalities

• Inequalities are statements that compare two expressions using >, <, ≥, ≤, or ≠.