Problem 7 - Prealgebra

Question

Graph each of the following ordered pairs. $$(-3,-4)$$

Step-by-Step Solution

Answer

Plot the point (-3,-4) by moving 3 left and 4 down from the origin.

Step 1: Understand the Coordinate Plane

The coordinate plane is divided into four quadrants. The horizontal line is the x-axis, and the vertical line is the y-axis. The center point, where the axes intersect, is called the origin (0,0). The first number in an ordered pair is the x-coordinate, and the second number is the y-coordinate.

Step 2: Identify the Components of the Ordered Pair

The ordered pair is (-3,-4). Here, -3 is the x-coordinate, which tells us how far left or right to move from the origin. -4 is the y-coordinate, which tells us how far up or down to move from the origin.

Step 3: Move Horizontally on the x-axis

Start at the origin (0,0). Since the x-coordinate is -3, move three units to the left on the x-axis because -3 is negative.

Step 4: Move Vertically on the y-axis

From your current position after moving left, look at the y-coordinate, which is -4. Move four units down because -4 is negative.

Step 5: Plot the Point

Once you have moved according to both the x-coordinate and y-coordinate, place a dot or mark at that final location. This point represents the ordered pair (-3,-4) on the graph.

Key Concepts

Understanding Ordered Pairs

Ordered pairs are a fundamental concept in graphing and geometry. An ordered pair, such as (-3,-4), consists of two numbers written in a specific sequence within parentheses. The first number, known as the x-coordinate, tells you how far to move horizontally on the coordinate plane: left if it's negative, right if it's positive. The second number, the y-coordinate, guides your vertical movement: Down if it's negative and up if it's positive.

Ordered pairs are always written in a particular order, hence the name "ordered." The x-coordinate always comes before the y-coordinate.

• Example: In (-3,-4), -3 is the x-coordinate, and -4 is the y-coordinate.

Grasping this concept is essential because it tells us exactly where to locate points on the coordinate system. Without this sequential arrangement, finding a precise position on the plane would be challenging.

Exploring the Coordinate System

The coordinate system is like a giant map that helps us find points easily. It's composed of two main lines: the x-axis and the y-axis. These axes intersect at a point called the origin, denoted as (0,0). The plane is divided into four sections, called quadrants.

In the coordinate system:

• The x-axis is the horizontal line that runs left to right.
• The y-axis is the vertical line that runs up and down.
• The origin is at the center, which is (0,0).

Understanding this setup helps us understand how to move around the coordinate plane using ordered pairs. Each quadrant has its own characteristics, such as the type of numbers (positive or negative) you'll find in each. Quadrant I contains positive x and y values, while Quadrant III, where our point (-3,-4) lies, contains negative x and y values.

Graphing Points on the Coordinate Plane

Graphing points, like plotting (-3,-4) on a coordinate plane, requires careful steps based on the ordered pair. First, you start at the origin (0,0). From there, look at the first number in your ordered pair, which is the x-coordinate. For (-3,-4), -3 tells us to move 3 units left along the x-axis.

Once you've moved horizontally, turn your attention to the y-coordinate. The second number, -4, instructs us to go 4 units down the y-axis.

• Begin at the origin.
• Move according to the x-coordinate.
• Adjust according to the y-coordinate.

Once you've completed moving based on both coordinates, place a mark or a dot at your final location. This is the graphical representation of the ordered pair. Practicing these steps ensures accurate graphing and a better understanding of spatial relationships on the coordinate plane.