Algebraic Equations and Factoring

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically written in the form . Solutions can be found using factoring, completing the square, or the quadratic formula.

• Quadratic Formula:

• Factoring: Express the quadratic as a product of two binomials and set each factor to zero.

• Example: Solve using the quadratic formula.

Polynomial Factoring

Factoring polynomials involves expressing them as products of lower-degree polynomials. Techniques include grouping, using special products, and long division.

• Grouping: Rearranging terms to factor by grouping common factors.

• Long Division: Used for dividing polynomials and finding factors.

• Example: Factor by long division.

Functions and Their Properties

Function Notation and Evaluation

A function assigns each input to a unique output. Evaluating a function means substituting a value for .

• Example: Evaluate and for a given piecewise function.

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

• Example: Find the domain of .

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• Example:

Graphing and Parent Functions

Graphing Functions

Graphing involves plotting points and analyzing features such as intercepts, asymptotes, and transformations.

• Parent Functions: Basic functions such as , , , .

• Transformations: Shifts, stretches, and reflections applied to parent functions.

• Example: Identify the parent function from a given graph.

Amplitude and Period of Trigonometric Functions

For functions like , the amplitude is and the period is .

• Example: Find amplitude and period for .

Trigonometric Functions and Identities

Evaluating Trigonometric Functions

Trigonometric functions relate angles to ratios of sides in right triangles. Common functions include sine, cosine, tangent, cosecant, secant, and cotangent.

• Example: Evaluate .

Trigonometric Table

Values of trigonometric functions for specific angles are often summarized in tables.

Additional info: Table entries inferred for standard angles.

Linear Equations and Systems

Equations of Lines

The equation of a line can be written in slope-intercept form or point-slope form . Lines perpendicular to a given line have slopes that are negative reciprocals.

• Example: Find the equation of a line passing through and perpendicular to .

Solving Systems of Equations

Systems of equations can be solved by substitution, elimination, or graphing. The solution is the point(s) where the equations intersect.

• Example: Solve and .

Function Transformations and Operations

Function Operations

Functions can be added, subtracted, multiplied, divided, and composed. The difference quotient is a key concept for understanding rates of change.

• Difference Quotient: For , simplifies to .

Set Notation and Intervals

Interval Notation

Intervals describe subsets of the real numbers. Use parentheses for open intervals and brackets for closed intervals.

• Example:

Exponent Rules and Simplification

Exponent Properties

Exponent rules are used to simplify expressions involving powers.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Example: Simplify and .

Factoring and Simplification of Expressions

Factoring by Grouping

Factoring by grouping is a method used when a polynomial has four or more terms. Group terms to factor common elements.

• Example: Factor by grouping.

Function Identification from Graphs

Piecewise and Absolute Value Functions

Some graphs represent piecewise or absolute value functions. Recognizing these from their shape is important for function identification.

• Example: Identify from a graph.

Summary Table: Parent Functions

Additional info: Table entries inferred for standard parent functions.