Linear Equations and Their Properties

Distance Between Two Points

The distance between two points and in the coordinate plane is given by the distance formula:

• Formula:

• Example:

Midpoint Formula

The midpoint of a segment connecting and is:

• Formula:

• Example:

Slope of a Line

The slope of a line passing through and is:

• Formula:

• Example:

Point-Slope Form of a Line

The equation of a line with slope passing through :

• Formula:

• Example:

Finding Intercepts

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

• Example: For , set : (x-intercept: ). Set : (y-intercept: ).

Equations of Circles

Standard Form of a Circle

The equation of a circle with center and radius :

• Formula:

• Example: (center , radius $2$)

Completing the Square

To write a general quadratic equation in standard form, complete the square:

• Example: becomes (center , radius $3$)

Functions and Their Properties

Definition of a Function

A function is a relation in which each input (x-value) maps to exactly one output (y-value).

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

• Domain and Range:

  • Domain (D): Set of all possible input values (x-values).

  • Range (R): Set of all possible output values (y-values).

  • Example: ,

Interval Notation

• Open Interval: excludes endpoints.

• Closed Interval: includes endpoints.

• Union:

Evaluating Functions

• Substitute: Replace with the given value.

• Example: If , then

Finding Intercepts of Functions

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

• Example: For , x-intercept: , y-intercept:

Function Transformations

Shifting and Stretching Graphs

Transformations change the position or shape of a graph:

• Vertical Shift: shifts up by units.

• Horizontal Shift: shifts left by units.

• Vertical Stretch: stretches by factor .

• Reflection: reflects over the x-axis.

• Example: shifted left 1, stretched vertically by 2, reflected over x-axis, and shifted up 3:

Symmetry of Functions

• Even Function: (symmetric about y-axis)

• Odd Function: (symmetric about origin)

• Example: is neither even nor odd

Quadratic Functions and Their Properties

Standard Form and Vertex

• Standard Form:

• Vertex:

• Example: For , vertex at

Finding Maximum and Minimum Values

• Local Minimum: Lowest point in a small region

• Absolute Minimum: Lowest point on the entire graph

• Example: For , absolute minimum at

Summary Table: Function Transformations

Additional info:

• Some examples and explanations were expanded for clarity and completeness.

• Graph sketches referenced in the notes were described in text for accessibility.