Basics of Graphing

Introduction to Graphs & The Coordinate System

Graphing is a foundational skill in College Algebra, involving the representation of equations and relationships on the rectangular coordinate system, also known as the Cartesian Plane.

• Rectangular Coordinate System (Cartesian Plane): Formed by two perpendicular number lines (axes) that intersect at the origin (0,0).

• Horizontal axis: x-axis

• Vertical axis: y-axis

• Ordered pairs (x, y): Position along axes in the form (x, y)

• Origin: The point (0, 0) where axes intersect

• Quadrants: The plane is divided into four quadrants, numbered counterclockwise starting from the upper right.

• x-values: Right of origin are positive, left are negative

• y-values: Above origin are positive, below are negative

Example: Plot the points A (1, 2), B (–2, 2), C (–3, –1), D (3, –4), E (–1, –3) on the graph. Example: Graph the points F (–2, –1), G (2, –2), H (2, 3), I (–2, 4). Identify the quadrant of each point. Additional info: Understanding the coordinate plane is essential for graphing equations and interpreting solutions visually.

Two Variable Equations

Many equations in algebra involve two variables, typically x and y, and their solutions are represented as points on the Cartesian plane.

• Equations with One Variable: Solutions are points on a 1D line (e.g., x = 2).

• Equations with Two Variables: Solutions are points (x, y) on a 2D plane (e.g., y = 2x + 1).

Example: Determine if points (1, 2), (–1, 0), (3, –1) satisfy the equation y = –x + 3 by substituting each pair into the equation. Additional info: If a point satisfies the equation, it lies on the graph; otherwise, it does not.

Graphing Two Variable Equations by Plotting Points

To graph an equation, substitute values for one variable and solve for the other to create ordered pairs, then plot these points.

1. Isolate y (or x) in the equation if possible.

2. Choose 3–5 values for x (or y).

3. Calculate corresponding y (or x) values.

4. Plot the (x, y) points and connect them with a smooth line.

Example: Graph the equation –2x + y = 1 by creating ordered pairs using x = –2, –1, 0, 1, 2. Practice: Graph y = x2 + 3 = 0 and y = √x + 1 by choosing points that satisfy the equation. (For the square root, choose positive x-values only.)

Graphing Intercepts

Intercepts are points where the graph crosses the x-axis or y-axis. These are useful for quickly sketching graphs and understanding the behavior of equations.

• x-intercept: The point(s) where the graph crosses the x-axis (y = 0).

• y-intercept: The point(s) where the graph crosses the y-axis (x = 0).

Example: Find the x- and y-intercepts of the graph shown. Example: Find the intercepts of the graph below and write the ordered pairs. Additional info: Intercepts are often used to check the accuracy of a graph and to solve equations graphically.

Summary Table: Steps for Graphing by Plotting Points

Key Terms

• Ordered Pair: A pair of numbers (x, y) that represents a point on the coordinate plane.

• Quadrant: One of the four regions into which the x- and y-axes divide the plane.

• Intercept: The point where a graph crosses an axis.

• Cartesian Plane: The plane defined by the x- and y-axes.