Problem 24 - Algebra 1: Concepts and Skills

Question

Without plotting the point, tell whether it is in Quadrant I, Quadrant II, Quadrant III, or Quadrant IV. $$ (3,9) $$

Step-by-Step Solution

Answer

The point (3,9) is in Quadrant I.

Step 1: Determine the Sign of the Coordinates

Analyze the given coordinates (3,9). Here, x-coordinate is 3, and y-coordinate is 9. In both cases, the coordinates are positive, as they are greater than zero.

Step 2: Match with Quadrantal System

Given that both coordinates are positive, refer to the description of the quadrantal system. As per the system, if both x and y are positive, the point is in Quadrant I.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graphically represent points using ordered pairs. An ordered pair consists of an x-coordinate and a y-coordinate, which we generally write as \(x, y\). This system helps to specify the location of a point on the plane. The horizontal line is called the x-axis, and the vertical line is called the y-axis. These axes intersect at the origin, which has coordinates \(0,0\). Points on this plane help in visualizing mathematical equations and geometric concepts.
Understanding the coordinate plane is a fundamental skill:

• The x-axis runs left to right, while the y-axis runs up and down.
• The plane is divided into four regions called quadrants.
• Each point on the plane is unique, depending on its coordinates.

Recognizing how the axes divide the plane is crucial in using the quadrantal system to locate points.

Quadrantal System

The quadrantal system arranges the coordinate plane into four sections, called quadrants. These quadrants are divided based on the signs of the coordinates: positive or negative.

• Quadrant I: Both x and y coordinates are positive. This is the top-right section of the plane.
• Quadrant II: The x-coordinate is negative, and the y-coordinate is positive. This is the top-left section.
• Quadrant III: Both coordinates are negative. This section lies at the bottom-left.
• Quadrant IV: The x-coordinate is positive, and the y-coordinate is negative. This is the bottom-right section.

Use this system strategically to determine the location of points like (3,9). Since both numbers are positive, it falls in Quadrant I. Remembering the order and the signs of the quadrants makes identifying points much easier.

Positive Coordinates

When we talk about positive coordinates, we mean both the x and y components of the ordered pair are greater than zero. Positive coordinates are essential when determining which quadrant a point is in on the coordinate plane. To locate a point with positive coordinates like (3,9):

• First, examine the x-value: a positive number means you move to the right of the origin along the x-axis.
• Next, examine the y-value: a positive number means you move upward from the origin along the y-axis.

Points with positive coordinates are always in Quadrant I because they are both to the right of the y-axis and above the x-axis. Mastering the concept of coordinates and their signs empowers you to confidently identify points and understand their positions in relation to the origin on the coordinate plane.