Problem 26 - Algebra 1

Question

Without plotting the point, tell whether it is in Quadrant I, Quadrant II, Quadrant III, or Quadrant IV. $$(-5,6)$$

Step-by-Step Solution

Answer

The point (-5,6) is located in Quadrant II.

Step 1: Identify the x-coordinate

The x-coordinate of the given point is -5. Negative values for x indicate that the point is located to the left of the origin. This narrows down our possibilities to Quadrant II and Quadrant III, where x-values are negative.

Step 2: Identify the y-coordinate

The y-coordinate of the given point is 6. Positive values for y indicate that the point is located above the origin. This means the point is located in Quadrant II, where y-values are also positive.

Step 3: Determine the Quadrant

Given that the x-coordinate is negative and the y-coordinate is positive, we can conclude that the point (-5,6) is located in Quadrant II.

Key Concepts

quadrants

In a coordinate plane, quadrants help us locate points based on their positions relative to the origin, which is where the x-axis and y-axis intersect. Each quadrant is like a section of the plane, separated by these axes.
There are four quadrants:

• Quadrant I: Both x and y coordinates are positive. This means any point here is up and to the right of the origin.
• Quadrant II: The x-coordinate is negative, while the y-coordinate is positive. Points in this quadrant are up and to the left of the origin.
• Quadrant III: Both coordinates are negative, placing points down and to the left of the origin.
• Quadrant IV: The x-coordinate is positive, and the y-coordinate is negative. This quadrant contains points down and to the right of the origin.

Understanding which quadrant a point is in helps in graphing and analyzing that point's behavior within a coordinate plane.

x-coordinate

The x-coordinate, or abscissa, is the first in the ordered pair \(x, y\), and it tells us how far left or right a point is from the origin. A positive x-value places the point to the right of the origin, while a negative x-value places it to the left.
When analyzing the point \((-5, 6)\), the x-coordinate is -5. This indicates a position to the left of the y-axis.
The importance of the x-coordinate lies in its ability to help identify which side of the coordinate plane the point resides in. It immediately helps us differentiate between Quadrants I and IV, which have positive x-coordinates, and Quadrants II and III, both with negative values.

y-coordinate

The y-coordinate, also known as the ordinate, is the second value in the ordered pair \(x, y\). It represents how high or low a point is from the origin. Positive y-values position the point above the x-axis, while negative y-values place it below.
For the point \((-5, 6)\), the y-coordinate is 6, meaning the point is above the x-axis.
Understanding the y-coordinate is crucial. It assists in pinpointing whether the point is located in Quadrants I or II, where y-coordinates are positive, or in Quadrants III or IV, where they are negative. Combining both x and y coordinates helps precisely determine the location of any point on the coordinate plane.