Problem 2 - Precalculus

Question

The point of intersection of the-and-axes is the ________, and the two axes divide the coordinate plane into four parts called ________.

Step-by-Step Solution

Answer

The point of intersection of the x and y axes is the origin, and the two axes divide the coordinate plane into four parts called quadrants.

Step 1: Identify the point of intersection

The point where the x-axis and y-axis intersect each other is the origin. It is the central point of the whole coordinate plane and is often denoted by the coordinates (0,0).

Step 2: Identify the divisions of the plane

The x and y-axes split the plane into four quadrants. These parts are called 'quadrants'. Starting from the upper right quadrant and moving counter-clockwise, they are named First Quadrant, Second Quadrant, Third Quadrant, and Fourth Quadrant.

Key Concepts

Origin

The origin is a fundamental concept in the geometry of the coordinate plane. It represents the point of intersection between the x-axis and the y-axis. This central point is the starting reference for locating any other point on the plane.
Located at (0,0), the origin serves as the coordinate plane's absolute center. In this position:

• The x-coordinate is 0, which signifies that the point is neither to the left nor the right of the vertical axis, known as the y-axis.
• The y-coordinate is 0, indicating that the point is neither above nor below the horizontal axis, known as the x-axis.

Understanding the origin is vital because it acts as a reference for determining the position and direction of other points. Moving along each axis starts from this pivotal spot.

Quadrants

The coordinate plane is neatly divided into four sections or parts called quadrants. These quadrants are created by the intersection of the x-axis and the y-axis at the origin (0,0). This division helps in identifying the precise location and sign of the coordinates for any given point.
The quadrants are named in a counter-clockwise direction starting from the upper right portion of the plane:

• First Quadrant: Here, both the x and y-coordinates are positive. It represents points to the right of the y-axis and above the x-axis.
• Second Quadrant: In this quadrant, the x-coordinates are negative, and the y-coordinates are positive. It holds points that are to the left of the y-axis and above the x-axis.
• Third Quadrant: This quadrant contains points where both coordinates are negative, indicating a position to the left of the y-axis and below the x-axis.
• Fourth Quadrant: The x-coordinates are positive while the y-coordinates are negative. Points here are to the right of the y-axis and below the x-axis.

Understanding these quadrants is crucial for locating points and describing movements or reflections on the coordinate plane.

X-axis and Y-axis

The x-axis and y-axis are the two essential lines that form the backbone of the coordinate plane. They intersect at the origin to create a framework for determining the position of points.
Let's take a closer look at each:

• X-axis: This is the horizontal line on the coordinate plane. It extends indefinitely in both positive (right) and negative (left) directions. The x-coordinate of any point on this line is its position along the x-axis.
• Y-axis: This is the vertical line that also extends infinitely upwards and downwards. The y-coordinate of a point is its position on this axis, determining how far up (positive) or down (negative) it is.

The axes help us understand and navigate through the plane by forming a grid system. By utilizing the x and y-values, we can locate any point within this system based on its distance from both the x-axis and the y-axis. This system is powerful for solving geometrical problems, plotting graphs, and modeling data visually.