Problem 19 - Algebra 1: Concepts and Skills

Question

Without plotting the point, tell whether it is in Quadrant I, Quadrant II, Quadrant III, or Quadrant IV. $$ (5,-3) $$

Step-by-Step Solution

Answer

The point (5, -3) is in Quadrant IV

Step 1: Identify the coordinates

The point given is (5, -3). This means the x-coordinate is 5 and the y-coordinate is -3.

Step 2: Determine the Quadrant

Since the x-coordinate (5) is positive and the y-coordinate (-3) is negative, this point is in the quadrant with positive x and negative y. This is Quadrant IV.

Key Concepts

Coordinate System

A coordinate system is a framework that allows us to determine the position of a point in space relative to certain reference points. The most common coordinate system in two dimensions is the Cartesian or rectangular coordinate system. It consists of two lines intersecting at a right angle, forming an 'L' shape. These lines are known as axes: the horizontal axis is the x-axis and the vertical axis is the y-axis.

Each axis is labeled with positive and negative numbers. The point where both axes intersect is called the origin, represented by the coordinates \(0,0\). The location of any point on this plane is given as an ordered pair \(x, y\), where \(x\) represents the horizontal displacement from the origin, and \(y\) represents the vertical displacement. Understanding the coordinate system is crucial as it is the foundation for identifying quadrants and plotting points.

Identifying Quadrants

Dividing the coordinate plane using the x and y axes results in four sections called quadrants. Each quadrant corresponds to a unique combination of positive and negative values for x and y coordinates.

In Quadrant I, both x and y coordinates are positive. Quadrant II contains points where the x coordinate is negative but the y coordinate is positive. Moving to Quadrant III, both coordinates are negative. Lastly, Quadrant IV has a positive x coordinate and a negative y coordinate. Identifying which quadrant a point falls into is a matter of looking at the signs of the coordinates. For example, the point \(5, -3\) is in Quadrant IV because 5 is positive and -3 is negative.

Plotting Points in a Coordinate Plane

The skill of plotting points in a coordinate plane is fundamental for visualizing and understanding mathematical concepts. To plot a point, we first locate the x coordinate on the horizontal axis, then find the y coordinate on the vertical axis, and where these two values meet in space is the location of our point.

For instance, to plot the point \(5, -3\), you would start at the origin \(0, 0\), move 5 units to the right since the x coordinate is positive, and then 3 units down because the y coordinate is negative. It's an important exercise that reinforces the relationship between algebraic expressions and geometric representation on the plane. Plotting points accurately can help in graphing functions, finding intersections, and understanding spatial relations among mathematical elements.