Problem 25 - 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. $$(0,3)$$

Step-by-Step Solution

Answer

The point (0,3) lies on the y-axis.

Step 1 - Identify the Coordinates

Look at the coordinates provided. The point given is \text($$0,3$$).

Step 2 - Analyze the x-coordinate

Check the x-coordinate of the point. If the x-coordinate is 0, the point lies on the y-axis. In the given point \text($$0,3$$), the x-coordinate is 0.

Step 3 - Conclude the Position

Since the x-coordinate is 0, the point lies on the y-axis. Therefore, the point \text($$0,3$$) is on the y-axis.

Key Concepts

quadrants

In the coordinate plane, we divide it into four sections called quadrants. These quadrants help us easily identify the position of any given point based on its coordinates. The coordinate plane is formed by two perpendicular number lines intersecting at a point called the origin.

The four quadrants are labeled as follows:

• Quadrant I: Both the x-coordinate and y-coordinate are positive.
• Quadrant II: The x-coordinate is negative, and the y-coordinate is positive.
• Quadrant III: Both the x-coordinate and y-coordinate are negative.
• Quadrant IV: The x-coordinate is positive, and the y-coordinate is negative.

It's a good practice to visualize these quadrants to understand where points will fall. Remember that if a point falls exactly on the x-axis or y-axis, it is not in any quadrant.

x-coordinate

The x-coordinate of a point tells us its horizontal position on the coordinate plane. This is the first number in a coordinate pair. To understand its significance, let's break it down:

• If the x-coordinate is positive, the point is to the right of the y-axis.
• If the x-coordinate is negative, the point is to the left of the y-axis.
• If the x-coordinate is zero, the point lies exactly on the y-axis.

For example, in the point \text($$0,3$$), the x-coordinate is 0. This immediately tells us that the point is on the y-axis. Understanding the role of the x-coordinate is crucial for determining the precise location of any point in the plane.

y-axis

The y-axis is the vertical number line in the coordinate plane. It is crucial for finding the vertical position of points. Here's how to understand its importance:

The y-axis divides the coordinate plane into its left and right halves.
Points on the y-axis will have an x-coordinate of 0.

• If the y-coordinate is positive, the point is above the x-axis.
• If the y-coordinate is negative, the point is below the x-axis.
• If the y-coordinate is zero, the point lies exactly on the x-axis.

In the exercise given, the point is \text($$0,3$$). Since the x-coordinate is 0, the point lies on the y-axis. The y-coordinate here is 3, meaning the point is 3 units above the x-axis. This detailed understanding helps us determine that \text($$0,3$$) is on the y-axis, precisely located 3 units above the origin.