Problem 25 - Algebra 1: Concepts and Skills

Question

Without plotting the point, tell whether it is in Quadrant I, Quadrant II, Quadrant III, or Quadrant IV. $$ (-5,-2) $$

Step-by-Step Solution

Answer

The point (-5,-2) is in Quadrant III.

Step 1: Identify the coordinates

Let the provided point be \(P(-5,-2)\). Here, -5 is the 'x' coordinate and -2 is the 'y' coordinate.

Step 2: Apply the quadrant rule

The rule for determining quadrant is based on the signs of x and y coordinates. In our case, both x and y are negative. Therefore, the point \(P(-5,-2)\) is in Quadrant III.

Step 3: Verify

In Quadrant III both x and y are negative. Here, x and y coordinates of point \(P(-5,-2)\) are also negative. Hence, our rule applied correctly.

Key Concepts

Coordinate Plane

The coordinate plane is a fundamental concept in geometry and algebra. Understanding it is crucial for identifying the location of points. It's essentially a two-dimensional plane created by two perpendicular lines:

• The horizontal line: Known as the x-axis.
• The vertical line: Known as the y-axis.

These axes intersect at a point called the origin. The origin is represented as (0,0) and divides the plane into four distinct sections, known as quadrants.
The coordinate plane allows us to plot and identify points using ordered pairs, usually written as \(x, y\). Here, "x" represents the horizontal position, and "y" designates the vertical position.
This system is extremely useful for visualizing relationships between algebraic equations and geometric shapes.

Quadrants

The coordinate plane is divided into four sections called quadrants. These quadrants are named using Roman numerals from I to IV, proceeding in an anti-clockwise direction:

• Quadrant I: Both x and y coordinates are positive. Points here are in the top-right section.
• Quadrant II: The x-coordinates are negative and the y-coordinates are positive. Points are located in the top-left section.
• Quadrant III: Both x and y coordinates are negative. Points are in the bottom-left section.
• Quadrant IV: The x-coordinates are positive and the y-coordinates are negative. This quadrant is in the bottom-right section.

Knowing which quadrant a point occupies helps in understanding its position relative to the origin. Quadrants also simplify the process of graphing equations or shapes on the coordinate plane and determining their properties.

Negative Coordinates

Negative coordinates occur when either the x, y, or both values in a point are less than zero. Understanding negative coordinates is vital to mastering the quadrant system on the coordinate plane.

• Negative x-coordinate: Positions the point to the left of the y-axis.
• Negative y-coordinate: Positions the point below the x-axis.

The point \( (-5,-2) \) is a great example to explore negative coordinates. Both numbers are less than zero, meaning the point lies in Quadrant III. This quadrant hosts all points where both x and y values are negative.
Grasping negative coordinates aids in correct placement of points and allows deeper understanding of graph-related problems, making manipulation of algebraic expressions and geometry simpler.