Special Products and Factorization Techniques

Quadratic Formula

The quadratic formula is used to solve quadratic equations of the form .

• Formula:

• Example: Solve

Special Products

Special algebraic products simplify the process of expanding and factoring expressions.

• Square of a Binomial:

• Product of Sum and Difference:

• Cube of a Binomial:

• Sum and Difference of Cubes:

Binomial Theorem

The binomial theorem provides a formula for expanding powers of binomials:

• Where is the binomial coefficient.

Factoring by Grouping

Factoring by grouping is a method used when a polynomial has four or more terms.

• Group terms to factor common factors, then factor out the common binomial.

• Example:

Functions

Notations and Definitions for a Cartesian Plane

The Cartesian plane is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Points are represented as ordered pairs .

• The point of intersection of the axes is called the origin .

• The axes divide the plane into four quadrants:

Function and Notations

A function is a relation between two variables such that to each value of the independent variable, there corresponds exactly one value of the dependent variable.

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Codomain: The set of all possible output values (y-values) that the function could possibly take.

• Range: The set of all actual output values (y-values) that the function takes for inputs from the domain.

• Functions can be represented by tables, graphs, or algebraic expressions.

Functions Specified by Algebraic Expressions

Not all equations in two variables define y as a function of x. For example:

• is a function (each x has one y).

• is not a function (a single x can correspond to multiple y values).

• is not a function (fails the vertical line test).

Vertical Line Test

A graph represents a function if and only if no vertical line intersects the graph at more than one point.

• Example: The graph of passes the vertical line test; does not.

Domain and Range of Functions

To find the domain and range of a function, consider the following:

• For even roots (e.g., square roots), the expression under the root must be non-negative.

• For denominators, the denominator must not be zero.

• For logarithmic functions, the argument must be positive.

Examples:

• For , domain:

• For , domain:

• For , domain:

Even and Odd Functions

Even functions: Satisfy for all in the domain. Their graphs are symmetric about the y-axis. Odd functions: Satisfy for all in the domain. Their graphs are symmetric about the origin. Neither: If neither condition is satisfied.

• Example (Even):

• Example (Odd):

• Example (Neither):

Piecewise-Defined Functions

A piecewise-defined function uses different formulas for different parts of its domain. For example, the absolute value function:

Increasing and Decreasing Functions

A function is increasing on an interval if whenever in the interval. It is decreasing if whenever .

Transformations of Functions

Transformations shift, stretch, compress, or reflect the graph of a function:

• : Shift up by units

• : Shift down by units

• : Shift left by units

• : Shift right by units

• : Vertical stretch by ()

• : Horizontal compression by ()

• : Reflect about the x-axis

• : Reflect about the y-axis

Operations on Functions

Functions can be added, subtracted, multiplied, or divided (except where the denominator is zero):

• ,

Composite Functions

The composite function is defined as . The domain of is the set of all in the domain of such that is in the domain of .

• Example: If and , then

Exponential Functions

Definition and Properties

An exponential function is a function of the form , where and .

• Properties:

  • Domain:

  • Range:

  • y-intercept:

  • Horizontal asymptote:

  • Continuous and one-to-one

  • Always increasing if ; always decreasing if

Properties of Exponents

Natural Exponential Function

The natural exponential function uses the base :

• Domain:

• Range:

Graph Transformations for Exponential Functions

• Vertical shifts:

• Horizontal shifts:

• Reflections: (about x-axis)

• Stretch/compression:

Summary Table: Exponential Function Characteristics

Additional info: These algebraic and function concepts are foundational for further study in mathematics, including calculus and applications in sciences such as chemistry and physics.