Problem 8 - 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

Always

Step 1: Understanding the Cartesian plane

In a 2-dimensional Cartesian plane, positions of all the points are determined by their x and y coordinates. Thus, for any point, the x-coordinate determines the horizontal distance and the y-coordinate determines the vertical distance from the origin.

Step 2: Understanding the x-axis

Points located on the x-axis have a unique property: their y-coordinate is always 0 because they are exactly on the horizontal line that goes through the origin. They don't have any vertical distance from the origin, hence their y-coordinate is 0.

Step 3: Matching to the given exercise

Relating to the given exercise, we can conclude that a point plotted on the x-axis always has a y-coordinate of 0.

Key Concepts

x-axis

In the realm of the Cartesian plane, the x-axis plays a crucial role. It is the horizontal line where all y-coordinates equal zero.
Imagine standing at sea level (x-axis) where there is no vertical movement.

• Points on the x-axis simply stretch horizontally left or right from the origin.
• This horizontal line is infinitely long in both positive and negative directions.
• Being on the x-axis signifies that a point is positioned neither above nor below it.

Understanding this concept is essential because any point that lies on the x-axis will always have its y-coordinate as zero.
As you view graphs or trudge through math problems, remember this: the x-axis serves as the starting line of horizontal navigation, without any rise or fall in altitude.

coordinates

Coordinates are the lifeblood of the Cartesian plane, allowing us to pinpoint a location precisely. Each point in this plane is described by a pair: the x-coordinate and the y-coordinate.

• The x-coordinate tells us how far to move horizontally from the origin. Positive values go right, negative go left.
• The y-coordinate indicates the vertical distance from the origin. Positive values move up, negative go down.

When you see a coordinate pair like (3, 0), that tells you to move three units right from the origin while staying level on the x-axis.
Each ordered pair of coordinates provides a unique address to a position on the grid, making it easy to locate points in mathematical and real-world contexts.

origin

The origin is the central point of the Cartesian plane, where the x-axis and y-axis intersect. It is represented by the coordinates (0,0).
This is the starting point for any adventure into the plane.

• At the origin, both the x-coordinate and y-coordinate are zero, which means no movement in any direction.
• It serves as a reference point for measuring all other positions in the plane.

This central spot helps us divide the plane into four quadrants, enabling a system to map out locations accurately.
Whenever you see the term 'origin,' think of it as the point of balance and symmetry from which everything else on the grid revolves around.