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A coordinate plane is a two-dimensional surface defined by two intersecting lines, known as axes. The horizontal axis is called the x-axis, and the vertical axis is called the y-axis. These axes divide the plane into four parts, known as quadrants, and they intersect at a point known as the origin, which has the coordinates (0, 0). Each point on the plane is represented as an ordered pair

• The first number represents the x-coordinate, which shows the point's horizontal position.
• The second number is the y-coordinate, indicating the point's vertical position.

In the given exercise, the points A(-3, 2) and B(-3, 5) are plotted on this plane. Here, both points share the same x-coordinate (-3). This reflects their positions to the left of the origin along the x-axis and different heights on the y-axis. Plotting these accurately on the coordinate plane is crucial for visualizing their alignment and the line they form.

A vertical line in geometry is a straight line that runs up and down the coordinate plane. Unlike horizontal lines that run left and right, vertical lines remain at a constant x-coordinate as you move along the line. In the exercise, both points A(-3, 2) and B(-3, 5) have the same x-coordinate, -3. This indicates that they form a vertical line. Characteristics of a vertical line include:

• It is parallel to the y-axis.
• It does not cross the x-axis except at the x-coordinate it is defined.
• It can extend infinitely in both directions along the y-axis.

Recognizing when a set of points forms a vertical line helps identify line properties and characteristics promptly.

The slope of a line is a measure indicating how steeply the line inclines or declines. Calculated using the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where

• \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of two distinct points on the line.

However, when both points have the same x-coordinate, as with A(-3, 2) and B(-3, 5), the denominator becomes zero:\[ x_2 - x_1 = (-3) - (-3) = 0 \]This results in dividing by zero, which is mathematically undefined. Thus, vertical lines have an undefined slope. It demonstrates the line's direction cannot be described in terms of rise over run because the run is zero, indicating a vertical slope. Understanding this concept is essential for graph and line analysis.

Plotting points on a coordinate plane is a foundational aspect of graphing lines and understanding geometric relationships. Each point is defined by an ordered pair \((x, y)\), and plotting these involves:

• Locating the x-coordinate on the x-axis.
• Identifying the y-coordinate on the y-axis.
• Marking the intersection of these values.

For example, to plot the points A(-3, 2) and B(-3, 5), we:

• Find -3 on the x-axis, move vertically to the y-coordinate 2 to mark A.
• Then again -3 on the x-axis, and up to 5 on the y-axis to mark B.

This precise plotting shows a clear visual representation of their location and helps identify the nature of the line connecting them. It is an essential technique for accurately depicting relationships in geometry.