Functions and Their Graphs

Linear, Quadratic, and Absolute Value Functions

Functions are mathematical relationships that assign each input exactly one output. Their graphs provide visual representations of these relationships.

• Linear Functions: The general form is , where m is the slope and b is the y-intercept.

• Quadratic Functions: The general form is . The graph is a parabola.

• Absolute Value Functions: The general form is . The graph is V-shaped.

• Transformations: Shifting, stretching, or reflecting graphs by changing parameters.

Example: The graph of is a parabola shifted up by 4 units.

Intercepts and Graph Analysis

Finding x- and y-intercepts

Intercepts are points where the graph crosses the axes.

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

Example: For , the y-intercept is and the x-intercept is .

Solving Linear Equations

Methods and Solutions

Linear equations can be solved by isolating the variable.

• Standard form:

• Solution:

Example: Solve :

• Expand:

• Add 24:

• Divide by 4:

Systems of Equations

Solving for Multiple Variables

Systems of equations involve finding values that satisfy all equations simultaneously.

• Substitution and elimination are common methods.

• Solutions may be unique, infinite, or nonexistent.

Example: Solve , , :

• Set

Formulas and Solving for Variables

Manipulating Formulas

Solving for a specified variable involves algebraic manipulation.

• Isolate the desired variable using inverse operations.

Example: Solve for .

Factoring and Expanding Expressions

Products and Standard Form

Factoring and expanding are used to simplify expressions and solve equations.

• Product:

• Standard form: Expand and combine like terms.

Example:

Rational Expressions

Division and Simplification

Dividing rational expressions involves multiplying by the reciprocal.

• Example:

Quadratic Equations

Solving by Formula and Completing the Square

Quadratic equations are solved using the quadratic formula or by completing the square.

• Quadratic formula:

• Completing the square: Rearranging to form

Example: Solve using the quadratic formula.

Discriminant and Nature of Solutions

Types of Roots

The discriminant determines the nature of the solutions of a quadratic equation.

• If : Two distinct real solutions

• If : One real solution

• If : Two complex solutions

Polynomial Equations and Zero Product Principle

Factoring and Solving

Polynomial equations can be solved by factoring and applying the zero product principle.

• Zero product principle: If , then or .

Example: factors to .

Inequalities and Interval Notation

Solving and Graphing

Inequalities are solved similarly to equations, but solutions are expressed as intervals.

• Interval notation: for all such that

• Graphing: Use number lines to represent solution sets

Example: Solve ; solution is , or .

Absolute Value Inequalities

Solving and Graphing

Absolute value inequalities are solved by considering both positive and negative cases.

• General form: leads to

Example: gives

Applications: Word Problems

Translating and Solving

Word problems require translating real-world scenarios into mathematical equations.

• Identify variables and relationships

• Set up equations and solve

Example: A fax machine charges $2.50 for the first page and nP = 0.65(n - 1) + 2.50$.

Summary Table: Types of Equations and Solution Methods

Additional info: Some context and explanations have been expanded for clarity and completeness, including general forms and solution steps for equations and inequalities.