Problem 23 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(21-32,\) indicate which quadrant contains the given point. If a point lies on one of the coordinate axes, indicate which one. $$(-2,-6)$$

Step-by-Step Solution

Answer

Quadrant III

Step 1: Understand Quadrants

In a Cartesian coordinate system, there are four quadrants. Quadrant I: (+,+), Quadrant II: (-,+), Quadrant III: (-,-), Quadrant IV: (+,-). Points on the axes do not belong to any quadrant but lie on the x-axis or y-axis.

Step 2: Locate the Point

The given point is (-2,-6). Identify the signs of the coordinates. Both x (=-2) and y (=-6) are negative.

Step 3: Determine the Quadrant

Points with both negative x and negative y coordinates lie in Quadrant III. The point (-2, -6) lies in Quadrant III.

Key Concepts

Quadrants in Geometry

In the Cartesian coordinate system, there are four distinct quadrants. This system consists of two perpendicular lines, called the x-axis and y-axis, which intersect at a point known as the origin \((0,0)\). The quadrants are numbered I through IV, moving counterclockwise from the positive x-axis.

Each quadrant holds different combinations of positive and negative coordinates:

• Quadrant I: Both coordinates are positive (\((+, +)\)).
• Quadrant II: The x-coordinate is negative, and the y-coordinate is positive (\((- , +)\)).
• Quadrant III: Both coordinates are negative (\((- , -)\)).
• Quadrant IV: The x-coordinate is positive, and the y-coordinate is negative (\((+, -)\)).

Understanding which quadrant a point belongs to helps in graphing and analyzing its position relative to the axes. Points on the axes do not belong to any quadrant.

Coordinate Axes

The coordinate axes are two perpendicular lines that intersect at the origin \((0,0)\). These axes divide the plane into the four quadrants. The horizontal line is known as the x-axis, and the vertical line is known as the y-axis.

The x-axis and y-axis help in uniquely locating any point in the plane using an ordered pair \((x,y)\). Here's what each axis represents:

• x-axis: Represents the horizontal value or 'abscissa'. Coordinates on the x-axis have a zero y-value \((x,0)\).
• y-axis: Represents the vertical value or 'ordinate'. Coordinates on the y-axis have a zero x-value \((0,y)\).

Understanding the coordinate axes is essential for reading and plotting points in the Cartesian plane. Points lying directly on these axes are described as being 'on the x-axis' or 'on the y-axis', and not in any of the four quadrants.

Negative Coordinates

Negative coordinates are crucial in identifying the position of a point in the Cartesian plane, particularly in Quadrants II and III.

When a point has a negative x-coordinate:

• It is situated to the left of the y-axis.

Similarly, when a point has a negative y-coordinate:

• It is situated below the x-axis.

If both the x and y coordinates are negative, like the given point \((-2, -6)\), it falls in Quadrant III.

For instance, in the example \((-2, -6)\):

• The x-coordinate is -2, indicating it's 2 units to the left of the origin.
• The y-coordinate is -6, indicating it's 6 units below the origin.

Understanding the role of negative coordinates is vital for correctly placing points and interpreting their positions on a graph.