Problem 39 - A Graphical Approach to Precalculus with Limits

Question

If the \(x\) -coordinate of a point is \(0,\) on which axis must the point lie?

Step-by-Step Solution

Answer

The point lies on the y-axis.

Step 1: Understanding the Problem

We are given that the \(x\)-coordinate of a point is \(0\). We need to find out on which axis the point must lie.

Step 2: Review Cartesian Coordinate System

In the Cartesian coordinate system, a point is represented as \((x, y)\). The \(x\)-axis is horizontal, and the \(y\)-axis is vertical. A point \((0, y)\) means the point has no horizontal displacement and lies on the \(y\)-axis.

Step 3: Determine the Axis

Since the \(x\)-coordinate is \(0\), the point has no displacement along the \(x\)-axis, meaning it's directly on the \(y\)-axis.

Step 4: Conclusion

Therefore, if the \(x\)-coordinate of a point is \(0\), the point must lie on the \(y\)-axis.

Key Concepts

Understanding the x-coordinate

In the Cartesian coordinate system, a point is typically defined by a pair of numbers written as \( (x, y) \). The first number of the pair is known as the x-coordinate. This number tells us the position of the point along the horizontal direction.

It is crucial to understand that the x-coordinate does not affect the movement of the point vertically. Instead, it indicates how far and in which direction the point is from the vertical axis, known as the y-axis, along the horizontal plane, which is the x-axis.

• A positive x-coordinate indicates a position to the right of the y-axis.
• A negative x-coordinate indicates a position to the left of the y-axis.
• An x-coordinate of zero means the point is exactly on the y-axis, with no horizontal movement.

Exploring the y-axis

The y-axis is one of the two principal axes in the Cartesian coordinate system. It runs vertically. This axis is crucial because it helps define the position of points in terms of height or depth above or below the x-axis, which is horizontal.

The y-axis is where the x-coordinate is zero, such that any point on this axis can be expressed as \( (0, y) \). Since there is no horizontal displacement, the position of the point will only vary depending on the value of the y-coordinate itself. Here's how it works:

• If the y-coordinate is positive, the point lies above the x-axis.
• If the y-coordinate is negative, the point is below the x-axis.
• At the origin, where both coordinates are zero, \( (0, 0) \), the point lies at the intersection of both axes.

Geometry and coordinate systems

Geometry is a branch of mathematics that deals with different shapes, sizes, and the properties of space. The Cartesian coordinate system is a crucial tool in geometry. It provides a clear framework for describing locations in a two-dimensional space using numerical coordinates.

This system divides the plane into four quadrants with the help of the x-axis and the y-axis, aiding in the graphical display of geometric figures and facilitating the understanding of geometric principles.

While dealing with geometric points, lines, and shapes:

• The Cartesian plane makes it easy to plot points and analyze the properties of shapes accurately.
• Coordinates allow for precise representation and calculation of distances and angles.
• Understanding the relationships between points and lines helps us solve complex geometric problems.

Grasping the concept of how points are plotted and located on the plane forms the foundation for more advanced studies in geometry.