Problem 20 - Algebra 1

Question

Without plotting the point, tell whether it is in Quadrant I, Quadrant II, Quadrant III, or Quadrant IV. $$(-2,7)$$

Step-by-Step Solution

Answer

The point (-2,7) lies in Quadrant II.

Step 1: Identify the given point and its coordinates

The given point is (-2,7). Therefore, the x-coordinate is -2, and the y-coordinate is 7.

Step 2: Apply the rules for quadrants

Since the x-coordinate is negative and the y-coordinate is positive, the point lies in Quadrant II.

Key Concepts

Understanding the Coordinate System

The coordinate system, also known as the Cartesian coordinate system, is a fundamental framework for geometry and algebra that allows us to pinpoint the location of a point in a 2-dimensional space. Imagine the system as a map that stretches endlessly in all directions, where each point is defined by a pair of numbers, these are the coordinates.

Breaking Down Coordinates

Coordinates are composed of two values: the x-coordinate and the y-coordinate. The x-coordinate represents a point's horizontal position while the y-coordinate represents its vertical position. The point where these axes intersect is called the origin, designated as (0,0).

• To move to the right of the origin, we increase the x-coordinate.
• To move to the left, we decrease it (using negative numbers).
• Moving up increases the y-coordinate.
• Moving down requires a decrease in the y-coordinate (again, using negative numbers).

The coordinate system is divided into four sections, called quadrants, each marked by Roman numerals I, II, III, and IV. They are counted in a counter-clockwise direction starting from the upper right quadrant.

Quadrant II Characteristics

In the context of the coordinate system, each quadrant exhibits unique characteristics based on the signs of the x and y coordinates. For Quadrant II, the features are distinct:

• The x-coordinate is always negative.
• The y-coordinate is always positive.

As a result, any point plotted in this quadrant will reflect those conditions. Think of Quadrant II as the upper-left section of the coordinate system. For a point like (-2, 7), you intuitively know it's in Quadrant II because the x-coordinate (-2) shows it's to the left of the origin, and the y-coordinate (7) shows it's above the origin.

Plotting Points

Plotting points is a crucial skill for visualizing and comprehending the relationships between different elements on the coordinate plane. Here’s a simplified process of plotting:

Step-by-Step Point Plotting

• Begin at the origin (0,0).
• Move horizontally to the right if the x-coordinate is positive, or to the left if it's negative.
• From that new position, move vertically up if the y-coordinate is positive, or down if it's negative.
• Mark the spot where the two movements converge – that's where you plot the point.

For the point (-2, 7), we move 2 units to the left and 7 units up from the origin. This simple approach negates the need for trial and error and provides a clear path to plot any point correctly.