Problem 28 - A Graphical Approach to College Algebra

Question

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(1,-4)$$

Step-by-Step Solution

Answer

The point (1, -4) lies in Quadrant IV of the coordinate system.

Step 1: Understanding the Coordinate System

The rectangular coordinate system is a plane divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). The quadrants are numbered counterclockwise starting from the upper right (Quadrant I), upper left (Quadrant II), lower left (Quadrant III), to the lower right (Quadrant IV). Each point is defined by an ordered pair (x, y), where x is the horizontal distance from the origin and y is the vertical distance from the origin.

Step 2: Identify the coordinates

Identify the x and y coordinates from the given point. For the point (1, -4), x is 1 and y is -4.

Step 3: Locate the point on the graph

Using the coordinate pair (1, -4), move 1 unit to the right on the x-axis (since x = 1) and 4 units down on the y-axis (since y = -4). Plot this point on the graph.

Step 4: Determine the quadrant

In the coordinate system, a positive x-coordinate and a negative y-coordinate places the point in Quadrant IV. This quadrant is located in the lower right section of the plane, where x > 0 and y < 0.

Key Concepts

Quadrants

The rectangular coordinate system is divided into four distinct sections, called quadrants. These quadrants help us easily identify the location and characteristics of any point in the system. They are labeled in a counterclockwise manner starting from the upper right section of the plane, which is Quadrant I.

• Quadrant I: Both x and y are positive, making this the region where (x, y) is positive-positive.
• Quadrant II: x is negative and y is positive, hence this is the negative-positive region.
• Quadrant III: Both x and y coordinates are negative, classifying it as the negative-negative region.
• Quadrant IV: x is positive and y is negative, placing it in the positive-negative region. This is where our given point (1, -4) lies.

Identifying the quadrant helps in visualizing the point’s location and understanding its relationship to the axes.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which we plot points with ordered pairs. It is formed by the intersection of two number lines: the horizontal x-axis and the vertical y-axis. These axes split the plane into four quadrants, as we've discussed. Whenever you plot a point on this plane, you first need to move along the x-axis by the x-coordinate and then along the y-axis for the y-coordinate. The intersection of these two movements gives us the location of our point. In the case of the point (1, -4), you start at the origin (0,0), move 1 unit to the right on the x-axis, and then 4 units down along the y-axis.
This motion accurately locates the point in the coordinate plane, demonstrating a simple yet powerful method to transform abstract numbers into a concrete spatial location.

Ordered Pairs

Every point on the coordinate plane is represented by an ordered pair (x, y), which tells you exactly where the point lies. The first number is the x-coordinate, indicating horizontal movement from the origin. The second number is the y-coordinate, which shows vertical movement. The order of these numbers is crucial; swapping them would result in a completely different location. For instance, the ordered pair (1, -4) directs you to move 1 step right and 4 steps down from the origin.
If it were (-4, 1), your movement would be to the left and then up, landing you in a different quadrant altogether. Understanding ordered pairs allows us to translate numerical data into spatial information, making it easier to engage with geometry visually. The systematic arrangement of numbers in (x, y) format simplifies complex graphing tasks into straightforward instructions.