Problem 3 - Prealgebra

Question

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

Step-by-Step Solution

Answer

Plot (-4, 2) by moving 4 units left and 2 units up on the coordinate plane.

Step 1: Understanding the Coordinate System

The coordinate system consists of two axes: the horizontal x-axis and the vertical y-axis. Points are plotted in this system with pairs of numbers, commonly called ordered pairs \( (x, y) \), where \( x \) corresponds to the horizontal position and \( y \) to the vertical position.

Step 2: Identifying the Ordered Pair

The given ordered pair is \( (-4, 2) \). This means that the point is located at \( x = -4 \) on the x-axis and \( y = 2 \) on the y-axis.

Step 3: Plotting the x-coordinate

From the origin (0,0) on the graph, move horizontally to the left because the x-coordinate is negative. Move 4 units left to reach \( x = -4 \).

Step 4: Plotting the y-coordinate

From \( x = -4 \), move vertically upwards because the y-coordinate is positive. Move 2 units up to reach \( y = 2 \).

Step 5: Marking the Point

After moving 4 units left and 2 units up, place a point at this position. Label it as \( (-4, 2) \) on the graph to indicate it represents the ordered pair.

Key Concepts

Coordinate System

The coordinate system is an essential concept in mathematics and is used to uniquely determine the position of points on a plane. It consists of two perpendicular lines, known as axes, that intersect at a point called the origin. This system allows for the precise plotting of points using pairs of numbers.

• The horizontal line is the x-axis.
• The vertical line is the y-axis.
• The point where these two lines intersect is called the origin, represented as \(0,0\).

The coordinate system is a powerful way to visualize mathematical concepts and is foundational for graphing ordered pairs.

X-axis and Y-axis

The x-axis and y-axis are the two main components of the coordinate system, acting as reference lines to determine positions in the plane.

The x-axis is horizontal and runs left to right. Points along this axis have a y-coordinate of zero, indicating no vertical displacement from the origin. It can be thought of as a number line.

The y-axis is vertical and runs bottom to top. Points along this axis have an x-coordinate of zero, showing no horizontal movement from the origin.

These axes divide the coordinate plane into four quadrants, each with distinct characteristics, helping to specify the location of a point based on its coordinates.

Plotting Points

Plotting points on the coordinate plane is a fundamental skill that involves placing a dot at a specific location, guided by an ordered pair of numbers. First, identify the x and y coordinates of your point.

• Start at the origin \(0, 0\).
• Move horizontally along the x-axis by the x-coordinate value: left if negative, right if positive.
• From that position, move vertically according to the y-coordinate value: down if negative, up if positive.
• Mark the point where you arrive and label it with its ordered pair.

Plotting is a step-by-step process that allows you to visually interpret data and mathematical expressions.

Ordered Pair (x, y)

An ordered pair, formatted as \(x, y\), is a set of two numbers used to represent a point on the coordinate plane. The order is crucial: the first number (\(x\)) refers to the horizontal position, and the second (\(y\)), the vertical.

For example, in the ordered pair \(-4, 2\), \(-4\) is the x-coordinate positioned four units left of the origin, while \(2\) is the y-coordinate, two units above the x-axis. The ordered pair uniquely identifies the point's location on the plane.

Understanding ordered pairs is vital in mathematics, as it forms the basis for graphing functions and solving geometric problems.