Problem 31 - 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. $$(489,-16)$$

Step-by-Step Solution

Answer

Quadrant IV

Step 1: Identify the coordinates

The given point is (489,-16). Here, 489 is the x-coordinate and -16 is the y-coordinate.

Step 2: Check the sign of the coordinates

To determine the quadrant for the point (489, -16), check the signs of the x and y coordinates. If the x-coordinate is positive and the y-coordinate is negative, the point lies in Quadrant IV.

Step 3: Point lies in Quadrant IV

Since the x-coordinate is 489 (positive) and the y-coordinate is -16 (negative), the point (489, -16) lies in Quadrant IV.

Key Concepts

Quadrants

In Coordinate Geometry, the coordinate plane is divided into four sections known as quadrants.
These quadrants help in identifying the position of points based on their coordinates.
Let's break down these quadrants:

• **Quadrant I**: Both x and y coordinates are positive (x > 0, y > 0)
• **Quadrant II**: The x-coordinate is negative, and the y-coordinate is positive (x < 0, y > 0)
• **Quadrant III**: Both x and y coordinates are negative (x < 0, y < 0)
• **Quadrant IV**: The x-coordinate is positive, and the y-coordinate is negative (x > 0, y < 0)

Each quadrant is crucial for determining the position and behavior of points in the coordinate plane. Understanding which quadrant a point belongs to can be helpful in graphing and analyzing mathematical functions.

Coordinate Axes

The coordinate plane is defined by two important lines: the x-axis and the y-axis. These axes intersect at a point called the origin, which is \( (0,0) \).

• **X-Axis**: The horizontal axis on the coordinate plane. Points on this axis have a y-coordinate of 0. For example, the point \( (3, 0) \) lies on the x-axis.
• **Y-Axis**: The vertical axis on the coordinate plane. Points on this axis have an x-coordinate of 0. For example, the point \( (0, -5) \) lies on the y-axis.

Understanding the role of these axes is fundamental when plotting points or interpreting their locations. If a point lies on either axis, it is said to be on the axis rather than in any quadrant.

Identifying Quadrants

To identify which quadrant a point lies in, look at the signs of its coordinates.

Let’s take the point \( (489, -16) \) from the exercise:

• The x-coordinate is 489, which is positive.
• The y-coordinate is -16, which is negative.

According to the quadrant rules:

• If the x-coordinate is positive and the y-coordinate is negative, the point lies in **Quadrant IV**.

Therefore, \( (489, -16) \) lies in Quadrant IV.
If both coordinates were positive, the point would be in Quadrant I.
Understanding how to identify the quadrant helps in graphing points accurately and interpreting their positions on the coordinate plane.