Graphing and Analyzing Linear Equations in the Rectangular Coordinate System

Graphs & the Rectangular Coordinate System

Introduction to the Cartesian Plane

The rectangular coordinate system, also known as the Cartesian Plane, is a two-dimensional plane formed by two perpendicular axes: the horizontal x-axis and the vertical y-axis. This system is fundamental for graphing equations and analyzing relationships between variables in precalculus.

• Horizontal axis: x-axis

• Vertical axis: y-axis

• Origin: The point (0, 0) where the x- and y-axes intersect

• Ordered pairs (x, y): Specify the position of points on the plane

• Quadrants: The axes divide the plane into four quadrants, numbered counterclockwise starting from the upper right (Q1)

Example: Plot the points A (4, 3), B (–3, 2), C (–2, –3), D (5, –4), E (0, 0), F (0, –3) on the graph.

Solving Two Variable Equations

Equations with One vs. Two Variables

In precalculus, equations may involve one or two variables. Understanding the difference is essential for graphing and analysis.

Equations with ONE Variable	Equations with TWO Variables
x + 2 = 5 x = 3	x + y = 5 y = 7 – x
Solution: point (x) on a 1D line	Solution(s): points (x, y) on a 2D plane

• To determine if a point (x, y) satisfies an equation, substitute x and y values into the equation.

• The graph of an equation visually represents all (x, y) pairs that make the equation true.

• If a point satisfies the equation, it lies on the graph; otherwise, it does not.

Example: For the equation x + y = 5, check if the points (3,2), (4,1), (0,5), (–1,6) satisfy the equation and plot them.

Graphing Two Variable Equations by Plotting Points

Step-by-Step Graphing Method

To graph an equation, calculate and plot ordered pairs (x, y) that satisfy the equation.

1. Isolate y to the left side: y = ...

2. Calculate y-values for 3–5 chosen x-values

3. Plot (x, y) points from Step 2

4. Connect points with a line or curve

Example: Graph the equation –2x + y = –1 by creating ordered pairs using x = –2, –1, 0, 1, 2.

x	y	Ordered pair (x, y)
–2
–1
0
1
2

Practice: Graph y – x2 + 3 = 0 and y = √x + 1 by choosing points that satisfy the equations.

Graphing Intercepts

Finding x- and y-Intercepts

Intercepts are points where a graph crosses the x-axis or y-axis.

x-Intercept	y-Intercept
x-value when graph crosses x-axis (y = 0)	y-value when graph crosses y-axis (x = 0)
Ordered pair: (x, 0)	Ordered pair: (0, y)

• To find x-intercepts, set y = 0 and solve for x.

• To find y-intercepts, set x = 0 and solve for y.

Example: Write the x- and y-intercepts of the given graph. Find the intercepts of another graph as practice.

Slopes of Lines

Definition and Calculation of Slope

The slope of a line measures how steep the line is. It is calculated as the ratio of the change in y to the change in x between two points on the line.

• Slope formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• Given two points (x1, y1) and (x2, y2), substitute their coordinates into the formula.

• The order of points does not affect the slope value.

Example: Find the slopes of lines A and B shown in the graph using the formula above.

Practice: Find the slope of the line shown, and the slope of the line containing the points (–1,1) and (4,3).

Summary Table: Key Concepts

Concept	Definition	Formula/Example
Ordered Pair	Coordinates specifying a point on the plane	(x, y)
Quadrant	One of four regions divided by axes	Q1, Q2, Q3, Q4
x-Intercept	Where graph crosses x-axis	Set y = 0, solve for x
y-Intercept	Where graph crosses y-axis	Set x = 0, solve for y
Slope	Steepness of a line	$m = \frac{y_2 - y_1}{x_2 - x_1}$

Additional info:

• These notes cover foundational graphing skills in precalculus, including plotting points, graphing equations, finding intercepts, and calculating slopes.

• Practice problems are included to reinforce each concept.