Problem 7 - Algebra 1: Concepts and Skills

Question

Complete the statement with always, sometimes, or never. A point plotted in Quadrant IV \({?}\) has a positive \(x\) -value.

Step-by-Step Solution

Answer

The correct term to complete the statement is 'always'. A point plotted in Quadrant IV always has a positive x-value.

Step 1: Understanding Quadrants in a Coordinate System

A plane is divided by two perpendicular lines, one vertical (y-axis) and one horizontal (x-axis), forming four quadrants. The characteristics of Quadrant IV are that it has positive x and negative y coordinates.

Step 2: Evaluate Statement

Now, with our understanding of Quadrants, we can evaluate the statement. In Quadrant IV, the x value is always positive.

Key Concepts

Perpendicular Lines

In a coordinate system, perpendicular lines play a crucial role. The two primary lines you will encounter are the x-axis and the y-axis. These lines intersect each other at right angles—meaning they are perpendicular. This intersection divides the plane into four sections known as quadrants. Each quadrant has unique characteristics based on the signs of the coordinates. Understanding perpendicular lines helps us analyze the coordinate plane efficiently. They essentially create a reference framework for locating points.

• The horizontal line is called the x-axis.
• The vertical line is called the y-axis.

Both lines meet at the origin, denoted as (0, 0). This origination point is the center of the quadrants, and from here each axis extends infinitely in two opposite directions.

x-axis and y-axis

The x-axis and y-axis are the two main components of a coordinate system. Each axis enables you to identify the location of a point by giving it a set of coordinates. The x-axis runs horizontally through the plane. It helps determine the horizontal position of a point.

• Points to the right of the y-axis have positive x-coordinates.
• Points to the left have negative x-coordinates.

The y-axis, on the other hand, is the vertical line. It determines the vertical position of a point:

• Points above the x-axis have positive y-coordinates.
• Points below have negative y-coordinates.

Together, these axes allow for a systematic way to specify locations on the plane, making it easier to communicate and calculate problems involving positions.

Positive and Negative Coordinates

Coordinates are numbers that describe the location of a point on the coordinate plane. Each point is represented by an ordered pair \(x, y\). Depending on the quadrant, these coordinates can be positive or negative.

• In Quadrant I: Both x and y are positive.
• In Quadrant II: x is negative, y is positive.
• In Quadrant III: Both x and y are negative.
• In Quadrant IV: x is positive, y is negative.

Quadrants are important because they allow us to make generalizations about the sign of coordinates in a particular region. For example, if a point is in Quadrant IV, we always know that the x-coordinate is positive and the y-coordinate is negative. This understanding helps in making more accurate predictions and evaluations about the behavior of points within each quadrant.