Problem 9 - Algebra 1: Concepts and Skills

Question

Complete the statement with always, sometimes, or never. A point plotted on the \(x\) -axis ? has \(y\) -coordinate \(0 .\)

Step-by-Step Solution

Answer

A point plotted on the x-axis always has a y-coordinate of 0

Step 1: Understanding the Cartesian Plane

In a Cartesian coordinate system, the x-axis is a horizontal line and the y-axis is a vertical line. They intersect at a point called the origin, where both x and y coordinates are 0.

Step 2: Coordinate of a Point on the x-axis

Any point on the x-axis has a y-coordinate of 0. This is due to the orientation of the axes where movement along the x-axis does not create any change in the y-coordinate.

Step 3: Finalizing the Answer

Given these characteristics, we can conclude that a point plotted on the x-axis always has a y-coordinate of 0. This is a characteristic feature of a Cartesian coordinate system.

Key Concepts

X-axis and Y-axis

Imagine you're standing at the center of a giant grid with lines running both vertically and horizontally. This is the essence of the Cartesian coordinate system - a fundamental framework for plotting points in a two-dimensional space.

In this system, the x-axis runs horizontally, like the horizon you see when gazing out across a calm sea. As you move to the right on this axis, the numbers increase positively. Conversely, moving to the left, the numbers become negative. Then, there's the y-axis, shooting up into the sky and burrowing down into the earth like an elevator. As you move up the y-axis, the numbers climb higher, and as you move down, they dip into negative values.

The axes meet at what's known as the origin, the point of reference for the entire system, like the 'home' position on this vast grid. It's a crucial component as it allows us to specify positions or locations within this two-dimensional framework.

Coordinates of a Point

When you're trying to tell a friend where you're standing in a park, you give them specific directions, like 'near the fountain' or 'by the oak tree'. In the Cartesian system, we give a precise numeric address to every speck on our grid - we call these the coordinates of a point.

A point's coordinates are like its home address, consisting of two numbers in parentheses. The first number is the x-coordinate, telling you how many steps left or right from the origin you'd need to take. The second is the y-coordinate, which tells you the number of steps up or down to take. Taken together, these coordinates will place you directly over the point's location on our grid. For example, (3, -2) would mean 3 steps to the right along the x-axis and 2 steps down along the y-axis.

Origin in Coordinate Plane

When we talk about the origin in a coordinate plane, we're referring to the starting point of our entire system. It's the 'zero point', where the journey along our axes begins. Think of it as the 'You Are Here' marker on a map of a shopping mall. In Cartesian coordinates, this is the point (0, 0), where the x-axis and the y-axis meet and cross each other.

The origin is critical in understanding the layout of the coordinate system. It not only separates the x-axis and y-axis into positive and negative sides but also acts as the reference point from which all other points are measured and plotted. With every other point on the plane, the distance from this central marker determines its exact location.