Fundamental Concepts of Algebra

Order of Operations

The order of operations is essential for evaluating mathematical expressions correctly. The standard order is Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right), often remembered by the acronym PEMDAS.

• Parentheses: Calculate expressions inside parentheses first.

• Exponents: Evaluate powers and roots next.

• Multiplication/Division: Perform these operations from left to right.

• Addition/Subtraction: Complete these last, also from left to right.

• Example:

Evaluating Expressions

Apply the order of operations to simplify algebraic expressions and solve equations.

• Combine like terms and use distributive property as needed.

• Example:

Equations and Inequalities

Solving Linear Equations

Linear equations can be solved by isolating the variable using inverse operations.

• Steps:

  1. Combine like terms on each side.

  2. Use addition/subtraction to isolate the variable term.

  3. Use multiplication/division to solve for the variable.

• Example:

Translating Word Problems

Translate verbal statements into algebraic equations using key words for operations.

Graphs

Graphing Linear Equations

Linear equations can be graphed using slope-intercept, point-slope, or standard form.

• Slope-Intercept Form: where is the slope and is the y-intercept.

• Point-Slope Form:

• Standard Form:

• Example: For , slope , y-intercept .

Vertical and Horizontal Lines

• Horizontal Line: (slope )

• Vertical Line: (undefined slope)

Functions & Graphs

Domain and Range

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Set Notation:

• Interval Notation: for included endpoints, for excluded endpoints.

• Example: For , domain , range .

Types of Numbers

Polynomial Functions

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of its factors.

• Greatest Common Factor (GCF): Factor out the largest common factor.

• Difference of Squares:

• Perfect Square Trinomials:

• Example:

Quadratic Functions

Standard, Vertex, and Intercept Forms

Quadratic functions can be written in several forms, each useful for different purposes.

• Standard Form:

• Vertex Form:

• Intercept Form:

• Axis of Symmetry:

• Vertex: in vertex form, or in standard form

Graphing Quadratic Functions

To graph a quadratic function, find the axis of symmetry, vertex, and plot additional points using symmetry.

• Opening Direction: If , opens up; if , opens down.

• Domain:

• Range: if opens up, if opens down

Maximum and Minimum Values

The maximum or minimum value of a quadratic function occurs at the vertex.

• Maximum: If , vertex is the maximum.

• Minimum: If , vertex is the minimum.

Applications: Projectile Motion

Quadratic Models in Physics

Projectile motion and falling objects can be modeled using quadratic functions.

• General Form: (height as a function of time)

• Vertex: Gives the maximum height and time to reach it.

• Example: For , vertex at sec, ft.

Exponent Rules

Properties of Exponents

Exponent rules are used to simplify expressions involving powers.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Zero Exponent: (for )

• Negative Exponent:

Set Notation and Interval Notation

Expressing Sets of Numbers

Set notation and interval notation are used to describe domains and ranges of functions.

• Set Notation:

• Interval Notation: , , , etc.

• Parentheses: Exclude endpoint; Brackets: Include endpoint.

Additional info:

• These notes cover foundational algebra, functions, graphing, and quadratic equations, which are essential for Precalculus and align with chapters 0-4, 19, and 20 of a standard Precalculus curriculum.