The Rectangular Coordinate System

Definition and Structure

The rectangular coordinate system, also known as the Cartesian plane, is a two-dimensional plane defined by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin (0, 0).

• Quadrants: The plane is divided into four regions, called quadrants, labeled I, II, III, and IV in a counterclockwise direction starting from the upper right.

• Ordered Pair: Each point on the plane is identified by an ordered pair (x, y), where x is the horizontal coordinate and y is the vertical coordinate.

X-Coordinate and Y-Coordinate

In an ordered pair (x, y):

• x-coordinate: The first value, representing the horizontal position.

• y-coordinate: The second value, representing the vertical position.

Key Formulas

• Distance Formula: The distance between points and is given by:

• Midpoint Formula: The midpoint of the segment connecting and is:

• Pythagorean Theorem: For a right triangle with legs of lengths a and b and hypotenuse c:

• Equation of a Circle (Standard Form): For a circle with center and radius :

Intercepts and Symmetry

Intercepts

• x-intercept: The point(s) where a graph crosses the x-axis. Set and solve for .

• y-intercept: The point(s) where a graph crosses the y-axis. Set and solve for .

Symmetry

• Y-axis Symmetry (Even Functions): A graph is symmetric with respect to the y-axis if replacing with yields the same equation. Example: .

• X-axis Symmetry: A graph is symmetric with respect to the x-axis if replacing with yields the same equation. Note: Graphs symmetric about the x-axis are not functions of .

• Origin Symmetry (Odd Functions): A graph is symmetric with respect to the origin if replacing with and with yields the same equation. Example: .

Graphing Equations and Functions

Basic Graphs

• Linear Functions: (diagonal line through the origin)

• Absolute Value: (V-shaped graph)

• Quadratic: (parabola opening upward)

• Cubic: (S-shaped curve)

• Square Root: (starts at origin, increases slowly)

• Reciprocal: (hyperbola, undefined at )

Transformations

• Vertical Shifts: shifts the graph up () or down ().

• Horizontal Shifts: shifts the graph right () or left ().

• Reflections: reflects across the x-axis; reflects across the y-axis.

• Stretching/Compressing: stretches vertically if , compresses if .

Linear Equations and Slope

Slope of a Line

• Definition: The slope measures the steepness of a line.

• Formula (Two Points):

• Slope-Intercept Form: , where is the slope and is the y-intercept.

• Point-Slope Form:

• Standard Form: , where and are not both zero.

Parallel and Perpendicular Lines

• Parallel Lines: Have equal slopes ().

• Perpendicular Lines: Slopes are negative reciprocals ().

Special Cases

• Vertical Lines: (undefined slope)

• Horizontal Lines: (slope = 0)

Functions and Their Properties

Definition of a Function

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

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

Function Notation

• Written as , where names the function and is the input variable.

• Example:

Even and Odd Functions

• Even Function: for all in the domain (symmetric about the y-axis).

• Odd Function: for all in the domain (symmetric about the origin).

Difference Quotient

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

• ,

Solving and Graphing Linear Functions

Steps to Analyze a Linear Function

1. Determine the slope and y-intercept.

2. Use the slope and y-intercept to graph the function.

3. Classify the function as increasing, decreasing, or constant based on the slope.

Example Table: Linear vs. Nonlinear Functions

This table represents a linear function because the change in y is constant for each change in x.

Applications and Word Problems

Linear Cost Functions

• Example: A phone company charges a flat fee plus a per-minute rate. The cost function is .

• To find the cost for a given number of minutes, substitute the value into the function.

• To find the number of minutes for a given cost, solve for .

Summary Table: Types of Symmetry

Additional info:

• Some example problems and graphs are included in the original material for practice. Students are encouraged to sketch graphs and solve for intercepts, symmetry, and transformations as part of their study routine.

• Difference quotient and function composition are foundational for later calculus topics.