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When you graph a linear equation like \(y = 3x - 1\), you do so on a coordinate plane. This plane has two perpendicular lines: the horizontal line is the x-axis, and the vertical line is the y-axis. These axes intersect at the origin point \(0, 0\). Each point on the plane is defined by a pair of numbers \(x, y\), known as coordinates.

The coordinate plane is divided into four quadrants, and it’s important to know how to navigate each. Positive x-values move you right from the origin, while negative x-values move you left. Similarly, positive y-values move you up, and negative y-values move you down. Being comfortable with these movements will make plotting points easier and more intuitive.

Understanding the layout and orientation of the coordinate plane is crucial for graphing any linear equation accurately. Choosing suitable x-values and correctly plotting them on the plane allows us to visualize the equation.

A table of values is a simple yet powerful tool used to set up the coordinates you'll plot. It helps organize your work when graphing the equation \(y = 3x - 1\).

Start by selecting some x-values. It's often easiest to choose numbers like -1, 0, and 1. Once you have these values, use the equation to calculate the corresponding y-values. For instance:

• If \(x = -1\), then \(y = 3(-1) - 1 = -4\).
• If \(x = 0\), then \(y = 3(0) - 1 = -1\).
• If \(x = 1\), then \(y = 3(1) - 1 = 2\).

This gives you the points \((-1, -4), (0, -1),\) and \(1, 2\).

The table of values acts as a guide for accurately plotting your points. It's a critical step that ensures your graph represents the equation correctly.

Plotting points is the final step in graphing a linear equation. Each \(x, y\) pair from your table of values becomes a specific point on the coordinate plane.

Start with the x-coordinate by moving horizontally along the x-axis. Then, move vertically to meet the y-coordinate.

• For the point \((-1, -4)\), move left to -1 on the x-axis, then down to -4 on the y-axis.
• For \((0, -1)\), stay at the origin and move down to -1.
• For \(1, 2)\), move right to 1, then up to 2.

As you plot the points, they should form a straight line. Draw a line through them, extending it across the plane to complete your graph.

Double-check that all points land on the line. This confirms the graph matches the equation, providing a visual representation of the relationship between x and y in \(y = 3x - 1\).