Review of Quadratic Equations

Factoring and Roots

Quadratic equations are polynomial equations of degree two, commonly written as . The solutions (roots) can be found by factoring or using the quadratic formula.

• Standard Form:

• Factored Form: , where , ,

• Quadratic Formula:

Example

• Solve using the quadratic formula.

Logarithmic Expressions

Definition and Properties

A logarithm is the inverse operation to exponentiation. The logarithm means .

• Definition:

• Properties:

Examples

Additional info: Logarithms with base are called natural logarithms, denoted .

Exponential Expressions

Properties of Exponents

Exponential expressions involve repeated multiplication of a base. The following properties are fundamental:

• Inverse Relationship:

Examples

• Solve

• Solve

Functions

Definition and Notation

A function from a set to assigns to each element in a unique real number .

• Domain: The set of all input values for which the function is defined.

• Range: The set of all possible output values of the function.

Example

• For , domain is , range is .

Substituting Values and Expressions

To evaluate a function, substitute the given value or expression for .

• Given , , ,

• For

Difference Quotient

The difference quotient is used to compute the average rate of change of a function:

• For ,

Linear Functions

Definition and Graph

A linear function has the form , where is the slope and is the y-intercept. Its graph is a straight line.

• Slope (): Measures the steepness of the line.

• Y-intercept (): The point where the line crosses the y-axis.

Equation of a Line

• General Form:

• Slope-Intercept Form:

• Vertical Line:

• Horizontal Line:

• Point-Slope Form:

Example

• Find the equation of the line passing through and : Slope: Equation: Substitute : Final equation: Or, using point-slope form:

Trigonometric Functions

Degrees and Radians Conversion

Angles can be measured in degrees or radians. To convert between them:

• Degrees to radians: multiply by

• Radians to degrees: multiply by

Unit Circle and Trigonometric Ratios

The unit circle helps define the sine, cosine, and tangent functions.

Graphs of Trigonometric Functions

• Sine Function: , domain , range

• Cosine Function: , domain , range

• Tangent Function: , domain , , range

Special Values

• , ,

• , ,

• , , does not exist (DNE)

Additional info: These foundational topics are essential for understanding calculus concepts such as limits, derivatives, and integrals.