Problem 1 - Prealgebra

Question

Graph each of the following ordered pairs. $$(4,2)$$

Step-by-Step Solution

Answer

Plot the point (4,2) 4 units right and 2 units up from the origin.

Step 1: Understand the Axes

First, identify the horizontal and vertical axes on the graph. The horizontal axis is called the x-axis, and the vertical axis is called the y-axis. This forms the coordinate plane where we will graph the ordered pair (4, 2).

Step 2: Locate the x-coordinate

Identify the first number in the ordered pair, which is 4. This tells us how far to move along the x-axis. Start at the origin (0,0), which is where the x- and y-axes intersect, and move 4 units to the right on the x-axis because the x-coordinate is positive.

Step 3: Locate the y-coordinate

Now identify the second number in the ordered pair, which is 2. Starting from the point (4,0) on the x-axis, move vertically up 2 units because the y-coordinate is positive.

Step 4: Mark the Point

After moving 4 units to the right along the x-axis and 2 units up along the y-axis, you should be at the point (4, 2). Place a dot or mark this point clearly on the graph, which represents the position of the ordered pair.

Key Concepts

Understanding the x-axis

The x-axis is the horizontal line running across the coordinate plane. It acts like a number line, where numbers increase as you move to the right and decrease as you move to the left.

When working with the coordinate plane, the x-axis is crucial in determining the horizontal position of a point. Consider an ordered pair \(x, y\). Here, the first number (\(x\)) tells you how many units to move left or right from the origin. The origin is where the x-axis and y-axis intersect, starting at \(0,0\).

• A positive x-coordinate instructs a move to the right.
• A negative x-coordinate means moving to the left.

For example, in the ordered pair \(4,2\), the x-coordinate is 4. Start at the origin and move 4 units right along the x-axis. Your new position is at (4,0), setting the stage for locating the y coordinate next.

Navigating the y-axis

While the x-axis deals with horizontal positioning, the y-axis handles vertical positioning. This vertical line runs up and down the coordinate plane. Numbers increase as you go up and decrease as you move down.

In an ordered pair \(x, y\), the second number (\(y\)) specifies how many units to move up or down from your current x-coordinate position. Once you're positioned on the x-axis based on your x-coordinate, the y-coordinate directs your movement vertically.

• A positive y-coordinate means moving upwards.
• A negative y-coordinate suggests moving downwards.

In our example, \(4,2\), once you reach the x-coordinate of 4, move up 2 units. This leads you to the final point of \[ (4, 2) \] on the plane.

Grasping Ordered Pairs

Ordered pairs are fundamental in locating points on the coordinate plane, composed of two numbers written in a specific order, usually within parentheses like \(x, y\). These pairs provide precise instructions on where to find a point:

• The first number tells how far to travel along the x-axis.
• The second number directs movement along the y-axis.

Consider the ordered pair given in the exercise, \(4, 2\). The key role of ordered pairs includes accuracy in plotting points. They eliminate confusion by clearly indicating the order and magnitude of movement in both horizontal and vertical directions.

This structure ensures that every point on the coordinate plane has a unique identity, highlighted by its specific coordinates.