Problem 22 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(21-32,\) indicate which quadrant contains the given point. If a point lies on one of the coordinate axes, indicate which one. $$(-2,6)$$

Step-by-Step Solution

Answer

The point \((-2,6)\) is in Quadrant II.

Step 1: Identify the Coordinates

The point given is \((-2,6)\). The first coordinate \((x)\) is \(-2\), and the second coordinate \((y)\) is \(6\).

Step 2: Determine the Sign of the Coordinates

Notice that the first coordinate \((x)\) is negative \(-2\) and the second coordinate \((y)\) is positive \((6)\).

Step 3: Determine the Quadrant

Using the signs of the coordinates: \(x < 0\) and \(y > 0\), we can determine that the point lies in the second quadrant.

Key Concepts

x and y coordinates

When working with coordinate geometry, we deal with points that have two values: the x-coordinate and the y-coordinate. The x-coordinate determines the position of the point along the horizontal axis. It tells us how far left or right the point is from the origin, which is the point where the x-axis and y-axis intersect.
The y-coordinate indicates the position along the vertical axis. It tells us how far up or down the point is from the origin. For example, in the point \((–2, 6)\), \(-2\) is the x-coordinate, and \(6\) is the y-coordinate. This means the point is 2 units to the left of the origin and 6 units up from the origin.
Remember:

• The x-coordinate comes first in a pair.
• The y-coordinate comes second.

Understanding these coordinates is essential for determining the position of points on the coordinate plane.

negative and positive coordinates

Coordinates can be either positive or negative, and this determines their position in relation to the origin.
If the x-coordinate is positive, it means the point is to the right of the origin. If it is negative, the point is to the left. Similarly, a positive y-coordinate means the point is above the origin, while a negative y-coordinate places it below.
Let’s break it down further:

• \(x > 0\): Point is to the right.
• \(x < 0\): Point is to the left.
• \(y > 0\): Point is above.
• \(y < 0\): Point is below.

For coordinates \((–2, 6)\), because the x-coordinate is negative and the y-coordinate is positive, the point is located to the left of the origin and above it. This understanding helps us place the point accurately within the four quadrants of the coordinate plane.

second quadrant

In coordinate geometry, the plane is divided into four quadrants. Each quadrant corresponds to a combination of positive and negative values for the x and y coordinates:
In the first quadrant, both coordinates are positive \((x > 0, y > 0)\).
In the second quadrant, the x-coordinate is negative, and the y-coordinate is positive \((x < 0, y > 0)\).
In the third quadrant, both coordinates are negative \((x < 0, y < 0)\).
In the fourth quadrant, the x-coordinate is positive, and the y-coordinate is negative \((x > 0, y < 0)\).
Points in the second quadrant, like \(–2, 6\), have a negative x-coordinate and a positive y-coordinate. This quadrant is located in the upper-left section of the coordinate plane.
Understanding the quadrant system helps in plotting points and visualizing their locations on the graph smoothly.