Graphs, Coordinate Systems, and Slopes in College Algebra

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Graphs & the Rectangular Coordinate System

Introduction to the Cartesian Plane

The rectangular coordinate system, also known as the Cartesian Plane, is fundamental in College Algebra for graphing equations and visualizing relationships between variables. It consists of two perpendicular axes that divide the plane into four quadrants.

Horizontal axis: The x-axis
Vertical axis: The 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 regions, labeled I, II, III, and IV, starting from the top-right and moving counterclockwise

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. Additional info: Quadrant I: (+x, +y), Quadrant II: (–x, +y), Quadrant III: (–x, –y), Quadrant IV: (+x, –y).

Identifying Quadrants

Each point (x, y) lies in a specific quadrant depending on the signs of its coordinates.

Quadrant I: x > 0, y > 0
Quadrant II: x < 0, y > 0
Quadrant III: x < 0, y < 0
Quadrant IV: x > 0, y < 0

Example: Graph the points W (1, –2), X (5, 2), Y (–3, –2), Z (–4, 3) and identify their quadrants.

Solving Two Variable Equations

Equations with One vs. Two Variables

Equations in algebra may involve one or two variables. Understanding the difference is crucial for graphing and solving equations.

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

Key Point: The graph of an equation is a visual representation of all (x, y) pairs that satisfy the equation.

If a point (x, y) satisfies the equation, it lies on the graph.
If it does not satisfy the equation, it does not lie on the graph.

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

Graphing Two Variable Equations by Plotting Points

Plotting Ordered Pairs

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

Isolate y to the left side: y = ...
Calculate y-values for 3–5 chosen x-values
Plot (x, y) points from Step 2
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

Additional info: Substitute each x-value into the equation to solve for y.

Practice Problems

Graph y – x2 + 3 = 0 by choosing points that satisfy the equation.
Graph y = √x + 1 by choosing positive x-values only.

x	y	Ordered Pair

Graphing Intercepts

x-Intercepts and y-Intercepts

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

x-intercept: The x-value when the graph crosses the x-axis. The y-value is always zero.
y-intercept: The y-value when the graph crosses the y-axis. The x-value is always zero.

x-Intercept	y-Intercept
x-value when y = 0	y-value when x = 0

Example: Write the x-intercepts and y-intercepts of the given graph.

Slopes of Lines

Definition and Calculation of Slope

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

Slope formula:
"Rise over run": Slope is the amount y changes for each unit x changes.
Order of points does not affect the slope value.

Example: Find the slope of Line A using points (x1, y1) and (x2, y2). Find the slope of Line B using points (2,4) and (1,2).

Practice Problems

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

Additional info: Positive slope: line rises left to right; negative slope: line falls left to right; zero slope: horizontal line; undefined slope: vertical line.

Summary Table: Key Concepts

Concept	Definition	Formula/Example
Ordered Pair	Location on the plane	(x, y)
Quadrant	Region of the plane	I, II, III, IV
x-intercept	Where graph crosses x-axis	y = 0
y-intercept	Where graph crosses y-axis	x = 0
Slope	Steepness of line

Additional info: These concepts form the foundation for graphing and analyzing linear equations in College Algebra.