College Algebra: Key Concepts, Functions, and Equations – Study Notes

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 $3 + 4i$ is $3 - 4i$.

Expressing Numbers as Fractions

To express a decimal as a fraction, write it as a ratio of integers and reduce to lowest terms.

• Example: $-2.312$ as a fraction is $-\frac{2312}{1000}$, 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 $(x_1, y_1)$ is:

$y - y_1 = m(x - x_1)$

• Key Point: Substitute the given point and gradient to find the specific equation.

• Example: For point $(1, -1)$ and gradient $\frac{3}{5}$, the equation is $y + 1 = \frac{3}{5}(x - 1)$.

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 ($A \cup C$): All elements in $A$, $C$, or both.

• Intersection ($A \cap C$): Elements common to both $A$ and $C$.

• Complement ($A'$): Elements in the universal set not in $A$.

• Example: If $A = \{-2, 10\}$ and $C = \{-4, 5\}$, then $A \cup C = \{-4, -2, 5, 10\}$.

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 $f(x) = 2x + 1$ is one-to-one.

Arrow Diagrams and Function Identification

Arrow diagrams visually represent functions by mapping elements from set $A$ (domain) to set $B$ (codomain).

• Key Point: If every element in $A$ maps to exactly one element in $B$, the diagram represents a function.

Quadratic Functions

General Form and Completing the Square

A quadratic function has the form $f(x) = ax^2 + bx + c$. Completing the square rewrites it as $a(x - h)^2 + k$.

• Formula: $f(x) = a(x + \frac{b}{2a})^2 + (c - \frac{b^2}{4a})$

• Example: For $f(x) = 2x^2 + 18x + 22$, complete the square:

$2x^2 + 18x + 22 = 2(x^2 + 9x) + 22 = 2(x^2 + 9x + (\frac{9}{2})^2) - 2(\frac{9}{2})^2 + 22$

$= 2(x + \frac{9}{2})^2 - 2(\frac{81}{4}) + 22 = 2(x + \frac{9}{2})^2 - 40.5 + 22 = 2(x + \frac{9}{2})^2 - 18.5$

Systems of Equations

Finding the Equation of a Line Through Two Points

Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope is $m = \frac{y_2 - y_1}{x_2 - x_1}$. The equation is then:

$y - y_1 = m(x - x_1)$

• Example: For points $(2, 3)$ and $(-1, 4)$, $m = \frac{4 - 3}{-1 - 2} = -\frac{1}{3}$.

Algebraic Expressions and Simplification

Simplifying Expressions

Combine like terms and use algebraic properties to simplify expressions such as $f(x) = x^2 - 10x + 12$.

• Example: Factor $x^2 - 10x + 12 = (x - 6)(x - 2)$.

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 $i$, where $i^2 = -1$.

• Power Set: The set of all subsets of a given set.

• The Factor Theorem: States that $x - a$ is a factor of a polynomial $f(x)$ if and only if $f(a) = 0$.