Problem 34 - A Graphical Approach to Precalculus with Limits

Question

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(-2,0)$$

Step-by-Step Solution

Answer

The point \((-2,0)\) lies on the x-axis, not in any quadrant.

Step 1: Understand the Coordinate System

A rectangular coordinate system is composed of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants.

Step 2: Identify the Point

The point given is \((-2, 0)\). This means it has an x-coordinate of \(-2\) and a y-coordinate of \(0\).

Step 3: Plot the Point

To plot \((-2, 0)\), move \(2\) units to the left from the origin (0,0) on the x-axis. Since the y-coordinate is \(0\), do not move vertically; the point remains on the x-axis.

Step 4: Determine the Location

Remember that the x-axis and y-axis are not part of any quadrant. Therefore, the point \((-2,0)\) is located on the x-axis and not in any quadrant.

Key Concepts

Quadrants

In the rectangular coordinate system, the two axes cross and divide the plane into four distinct areas, called quadrants. Each quadrant can be identified by the signs of the coordinates of the points located within it:

• The first quadrant is located in the upper right and contains points where both the x and y coordinates are positive.
• The second quadrant is in the upper left, where x is negative, and y is positive.
• The third quadrant is in the lower left, with both x and y being negative.
• The fourth quadrant is in the lower right, where x is positive, and y is negative.

Points located directly on one of the axes are not considered to be in any quadrant; they belong to the axes themselves. When determining the quadrant for any point, always look at the sign of the coordinates.

X-axis

The x-axis is a vital part of the rectangular coordinate system. It's the horizontal line that extends infinitely in both directions and serves as the baseline for measuring the x-coordinate.

• Positive x-coordinates are found to the right of the origin, while negative x-coordinates lie to the left.
• The x-axis is significant for determining where a point lies in terms of left-right positioning.
• Points lying directly on the x-axis will always have a y-coordinate of zero.

When a point, such as \((-2, 0)\), is located on the x-axis, it indicates no vertical movement from the origin. Thus, the y-coordinate is zero, and this point does not belong to any quadrant.

Y-axis

The y-axis is the vertical line in the rectangular coordinate system, intersecting the x-axis at the origin \(0, 0\). It is essential for measuring how far up or down a point is.

• Positive y-coordinates are above the origin, while negative y-coordinates are below.
• The y-axis acts as a reference line for how high or low each point is located.
• Any point on the y-axis will always have an x-coordinate of zero.

Although axis points are excluded from quadrants, the y-axis is crucial for helping to determine a point's vertical position. Knowing its y-coordinate helps one understand how that point might relate to the quadrants, but if a point has an x-coordinate equal to zero, it is on the y-axis, not within any quadrant.