Basics of Graphing

Introduction to Graphs & The Coordinate System

Graphing is a foundational skill in precalculus, involving the representation of equations and relationships on the rectangular coordinate system, also known as the Cartesian Plane. This system uses two perpendicular number lines (axes) to form a two-dimensional plane.

1. Horizontal axis: x-axis

2. Vertical axis: y-axis

3. Origin: The point (0, 0) where the axes intersect

4. Ordered pairs: Points are written as (x, y)

5. Quadrants: The plane is divided into four quadrants, numbered I to IV, starting in the upper right and moving counterclockwise

6. x-values: Indicate left/right position relative to the origin

7. y-values: Indicate above/below position relative to the origin

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

Two-Variable Equations

Many equations in precalculus involve two variables, typically x and y. These equations are represented as graphs in the coordinate 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)

To determine if a point (x, y) is a solution to an equation, substitute the values into the equation and check if it is true. If true, the point lies on the graph; if false, it does not.

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

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 (x, y) that satisfy the equation. Plot these points and connect them to reveal the graph.

• Choose several x-values

• Solve for corresponding y-values

• Plot the resulting (x, y) pairs

• Connect the points with a smooth curve or line

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

Steps for Graphing by Plotting Points:

1. Isolate y (or x) in the equation

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

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

4. Connect points with a line/curve

Practice: Graph the equation y = x2 + 3 = 0 by choosing points that satisfy the equation.

Practice: Graph the equation y = √x + 1 by choosing positive x-values only.

Graphing Intercepts

Intercepts are points where a graph crosses the x-axis or y-axis. These are important for understanding the behavior of graphs.

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

Example: Find the intercepts of the graph below.

If asked for "x- or y-intercepts," simply write the x- or y-value. If asked for just "intercepts," write the ordered pairs.

Additional info: These foundational graphing skills are essential for later topics in precalculus, including functions, equations, and analytic geometry.