Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

In which quadrants do the \(x\) -coordinates and the \(y\) -coordinates have the same sign?

Step-by-Step Solution

Answer

The \(x\) -coordinates and the \(y\) -coordinates have the same sign in Quadrant I and Quadrant III.

Step 1: Understand Quadrant I

In the first quadrant, both the \(x\) and the \(y\) coordinates hold positive values. So in Quadrant I, \(x\) -coordinates and the \(y\) -coordinates have the same sign, which is '+' (positive).

Step 2: Understand Quadrant II

In the second quadrant, the \(x\) coordinate is negative, and the \(y\) coordinate is positive. So in Quadrant II, \(x\) -coordinates and the \(y\) -coordinates have different signs.

Step 3: Understand Quadrant III

In the third quadrant, both the \(x\) and \(y\) coordinates are negative. So in Quadrant III, \(x\) -coordinates and the \(y\) -coordinates have the same sign, which is '-' (negative).

Step 4: Understand Quadrant IV

In the fourth quadrant, the \(x\) coordinate is positive, and the \(y\) coordinate is negative. So in Quadrant IV, \(x\) -coordinates and the \(y\) -coordinates have different signs.

Key Concepts

Quadrants

In coordinate geometry, the Cartesian plane is divided into four sections known as quadrants. Each quadrant is designated a specific sign combination based on the location of the points it contains.
The quadrants are numbered in a counterclockwise direction, starting from the upper right corner. It is important to know whether both the coordinates (x and y) have the same sign or different signs in each of these quadrants.

• Quadrant I: Here, both the x and y coordinates are positive, resulting in the combination (+/+). This means any point in this quadrant will have positive values for both the x and y coordinates.
• Quadrant II: In this section, x is negative while y is positive. Points in this quadrant are represented by the combination (-/+), indicating opposite signs for the coordinates.
• Quadrant III: This quadrant holds negative values for both x and y coordinates. The coordinates are (-/-). Therefore, any point here will have negative values for both x and y.
• Quadrant IV: Finally, this quadrant has x as positive and y as negative, indicated by the combination (+/-). Points will have opposite signs for the coordinates.

Coordinate Signs

Signs of the coordinates in a Cartesian plane determine the position of the points within the quadrants. Each quadrant hosts specific combinations of x and y signs.

• Positive/Positive (+/+): Both coordinates are above zero. This combination appears in Quadrant I.
• Negative/Positive (-/+): The x coordinate is below zero, while the y coordinate is above zero. These signs are found in Quadrant II.
• Negative/Negative (-/-): Both coordinates are below zero, occurring in Quadrant III. Points with these sign criteria will fall here.
• Positive/Negative (+/-): The x coordinate is above zero, while the y coordinate is below zero. This happens in Quadrant IV.

Understanding the signs of coordinates is crucial for graph interpretation and solving problems involving coordinate pairs.

Cartesian Plane

The Cartesian plane is a two-dimensional plane formed by the intersection of two perpendicular number lines, the horizontal x-axis and the vertical y-axis. This plane is pivotal in geometry for plotting points and analyzing shapes and graphs.

• X-axis: The horizontal line that divides the plane. Points on this axis have a y-coordinate of zero.
• Y-axis: The vertical line. Points here have an x-coordinate of zero.

The intersection point of the x-axis and y-axis is called the origin, noted as (0, 0). From this central point, you can map coordinates throughout the four quadrants. Each point or coordinate pair in the plane can be described as (x, y), where 'x' is the horizontal distance from the origin and 'y' is the vertical distance. For example, (3, 2) would be located in Quadrant I, as both numbers are positive. This spatial understanding is foundational for conducting geometric analyses, plotting graphs, and carrying out mathematical functions on the plane.