Problem 40 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The point lies on the x-axis.

Step 1: Understand the Coordinate System

In a 2D Cartesian coordinate system, every point is represented by an ordered pair \( (x, y) \). The first number \( x \) is the horizontal position, and the second number \( y \) is the vertical position.

Step 2: Condition Given in Problem

The problem states that the \( y \)-coordinate is \( 0 \). In terms of the coordinate system, this means the point is positioned in such a way that it does not move up or down from the origin along the \( y \)-axis.

Step 3: Determine the Axis for y=0

When \( y = 0 \), the point has no vertical displacement from the x-axis. Thus, all points where \( y = 0 \) lie along the horizontal x-axis. Therefore, the point must lie on the x-axis.

Key Concepts

x-axis

Imagine you're standing on a flat surface without moving up or down. This is similar to what the x-axis is in the Cartesian coordinate system. The Cartesian system is a plane with two intersecting lines: a horizontal one called the x-axis, and a vertical one called the y-axis.
On this plane, every position is determined by two numbers, forming a pair, like coordinates on a treasure map. When the second number of this pair, the y-coordinate, is zero, it shows that the point is neither above nor below the x-axis, making it reside exactly on it. It’s like walking on a straight line without climbing any hills or sliding into any valleys. All the action happens along the x-axis, the horizontal pathway.
Let’s think about how each point with a y-coordinate of zero will line up exactly on this horizontal stretch. Just like people lined up on a parade stand, none of them are in the air or underground—each stands precisely on the line of the street—the x-axis.

ordered pair

In the world of ordering groceries online, you might place an order by listing what you need and how much of each item. Similarly, in a 2D Cartesian coordinate system, positions are specified by an ordered pair, denoted (x, y) . In this pair, the first number, such as x, indicates a horizontal position on the x-axis, similar to how you decide whether to list bread before milk on your grocery list, because it’s arranged as first things-first. The second number, y, determines the vertical position on the y-axis.
Understanding ordered pairs is crucial for pinpointing exact locations on a coordinate plane. Think of each ordered pair as precise coordinates that help map out a space, just like using a compass on treasure island to find hidden riches based on specified directions.

y-coordinate equals zero

Picture a balloon tied to a string with its base on the ground so it doesn’t float up. In the Cartesian coordinate system, when a y-coordinate equals zero, it’s akin to keeping that balloon grounded. Let's explore this concept deeper:

• A point with a y-coordinate of zero is said to have no vertical movement. It stays exactly at the level where the y-axis intercepts the x-axis.
• Such a point has all its movement restricted to the horizontal x-axis, meaning visually, it’s grounded on the flat plane at no height above or below the horizontal line.
• The importance of having a y-coordinate zero helps us identify that the location purely depends on where it falls on the x-axis, like choosing a spot on a massive football field without ever moving up the stadium steps or going underground!

This principle clarifies that every point with their y-coordinate set to zero does not venture away from the horizontal line, emphasizing linear alignment along the x-axis.