Problem 1 - College Algebra

Question

Is it possible for a point plotted in the Cartesian coordinate system to not lie in one of the four quadrants? Explain.

Step-by-Step Solution

Answer

Points on the axes or at the origin do not lie in any quadrant.

Step 1: Understand the Quadrants

In a Cartesian coordinate system, the plane is divided into four quadrants by the x-axis and y-axis. These quadrants are labeled as Quadrant I (positive x and y), Quadrant II (negative x and positive y), Quadrant III (negative x and y), and Quadrant IV (positive x and negative y). Any point with coordinates (x, y) where both x and y are non-zero will fall into one of these quadrants.

Step 2: Identify Points not in Quadrants

To determine if a point does not lie in any quadrant, consider points that fall on the axes. Points on the x-axis have coordinates where y = 0 (e.g., (x, 0)), and points on the y-axis have coordinates where x = 0 (e.g., (0, y)). These points lie on the dividing lines and not in any specific quadrant.

Step 3: Consider the Origin

The origin, which is the point (0, 0), is a special case. It is the intersection of the x-axis and y-axis. Since the origin does not satisfy the conditions of being in any particular quadrant (positive or negative x and y), it also does not lie in any quadrant.

Step 4: Conclusion

It is possible for a point to not lie in any quadrant if it is on an axis (x-axis or y-axis) or at the origin. Such points are not enclosed entirely within the boundaries of the quadrants and thus are not considered to belong to any of them.

Key Concepts

Quadrants

In the Cartesian coordinate system, the plane is divided into four regions, known as quadrants. These quadrants are defined by the intersection of two perpendicular lines: the x-axis and the y-axis. Each quadrant is home to points with specific signs (positive or negative) for their x and y coordinates. Here's a quick breakdown:

• **Quadrant I**: Both x and y coordinates are positive (e.g., (3, 4)).
• **Quadrant II**: x is negative, while y is positive (e.g., (-2, 5)).
• **Quadrant III**: Both x and y coordinates are negative (e.g., (-4, -3)).
• **Quadrant IV**: x is positive, while y is negative (e.g., (6, -1)).

To determine the quadrant a point belongs to, simply look at the sign of its coordinates. Points that have x or y equal to zero do not lie in any quadrant because they fall directly on the axes, which are the boundaries separating the quadrants.

x-axis

The x-axis is a horizontal line that spans the Cartesian plane. It serves as a reference line for measuring the horizontal position of a point. When you see a point with coordinates (x, 0), it means that the point lies somewhere on the x-axis, regardless of whether x is positive or negative. In other words, its position is measured along the horizontal direction, with no vertical displacement.

Points on the x-axis:

• Have their y-coordinate equal to zero.
• Can be found in both positive and negative directions, stretching infinitely in both.
• Are not included within any quadrant, since they do not possess both a non-zero x and y coordinate simultaneously.

When considering points on the x-axis, remember that these points sit right on the line that divides Quadrants I and IV on one side, and Quadrants II and III on the other.

y-axis

The y-axis is the vertical counterpart to the x-axis and is crucial for determining the vertical position of a point. In the Cartesian coordinate system, any point that has coordinates of the form (0, y) will lie on the y-axis. This vertical line extends infinitely upwards and downwards, bisecting the plane into left and right halves.

Important facts about points on the y-axis:

• They possess an x-coordinate of zero.
• Only the y-coordinate offers information about their vertical position.
• These points are outside of the four quadrants, as they do not meet the criteria of having both x and y non-zero.

Points on the y-axis form the boundary between Quadrants I and II, and Quadrants III and IV. This essential axis helps in evaluating the symmetry and positioning of objects or functions on the Cartesian plane.

Origin

The origin in the Cartesian coordinate system is the central point where the x-axis and y-axis intersect. This unique point is denoted by the coordinates (0, 0) and acts as a reference for plotting other points on the plane. Because neither coordinate is positive or negative, the origin does not belong to any of the four quadrants. It holds a special position, marking the center of the coordinate plane.

Significance of the origin:

• It represents the "zero" point for both axes.
• All distance measurements on the plane derive from this point.
• It acts as a central pivot for rotational symmetry, especially in geometric transformations.

The origin is a crucial anchor in math and physics, as it often simplifies calculations and helps in visualizing geometric transformations. Understanding its role is fundamental to mastering the Cartesian coordinate system.