뒤로College Algebra: Key Concepts, Functions, and Equations – Study Notes
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자료에 맞춘 맞춤형 노트, 핵심 정의, 예시, 맥락을 확장해 제공합니다.
Complex Numbers and Their Properties
Conjugates of Complex Numbers
Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined by i2 = -1. The conjugate of a complex number a + bi is a - bi.
Key Point: The conjugate changes the sign of the imaginary part.
Example: The conjugate of is .
Expressing Numbers as Fractions
To express a decimal as a fraction, write it as a ratio of integers and reduce to lowest terms.
Example: as a fraction is , which can be simplified further.
Equations of Lines
Equation of a Line Given a Point and Gradient
The equation of a straight line with gradient (slope) m passing through point is:
Key Point: Substitute the given point and gradient to find the specific equation.
Example: For point and gradient , the equation is .
Sets and Set Operations
Universal Set, Subsets, and Set Notation
A universal set contains all elements under consideration. Subsets are sets whose elements all belong to the universal set.
Union (): All elements in , , or both.
Intersection (): Elements common to both and .
Complement (): Elements in the universal set not in .
Example: If and , then .
Functions and Their Properties
Definition of a Function
A function is a relation that assigns exactly one output to each input from a given set.
One-to-One Function: Each output is paired with only one input.
Example: The function is one-to-one.
Arrow Diagrams and Function Identification
Arrow diagrams visually represent functions by mapping elements from set (domain) to set (codomain).
Key Point: If every element in maps to exactly one element in , the diagram represents a function.
Quadratic Functions
General Form and Completing the Square
A quadratic function has the form . Completing the square rewrites it as .
Formula:
Example: For , complete the square:
Systems of Equations
Finding the Equation of a Line Through Two Points
Given two points and , the slope is . The equation is then:
Example: For points and , .
Algebraic Expressions and Simplification
Simplifying Expressions
Combine like terms and use algebraic properties to simplify expressions such as .
Example: Factor .
Key Definitions in Algebra
Relation: A set of ordered pairs, showing a relationship between elements of two sets.
One-to-One Function: A function where each output is paired with only one input.
Imaginary Set: The set of numbers involving , where .
Power Set: The set of all subsets of a given set.
The Factor Theorem: States that is a factor of a polynomial if and only if .
