Problem 86 - Algebra A Combined Function

Question

Write an ordered pair for each point described. Point \(D\) is three units to the left of the origin.

Step-by-Step Solution

Answer

Point D is at \((-3, 0)\).

Step 1: Understand the Origin

The origin on a coordinate plane is where the x-axis and y-axis intersect. It is represented by the ordered pair \((0, 0)\).

Step 2: Identify the Direction

The problem states that point \(D\) is three units to the left of the origin. On a coordinate plane, moving to the left from the origin corresponds to moving in the negative direction along the x-axis.

Step 3: Calculate the New Coordinates

Starting from the origin \((0, 0)\), move three units left on the x-axis. This changes the x-coordinate to \(-3\), while the y-coordinate remains 0.

Step 4: Write the Ordered Pair

The new location of point \(D\) after moving three units to the left of the origin is \((-3, 0)\).

Key Concepts

ordered pair

An ordered pair is a fundamental concept when working with the coordinate plane. It consists of two elements, typically expressed as

• The first element which represents a position along the x-axis.
• The second element which gives the position on the y-axis.

The format is usually written as (x, y), reflecting these two components. This notation is essential for specifying exact locations within the plane. In the context of our exercise, point D is represented by the ordered pair o(-3, 0) , with o-3 denoting its position on the x-axis three units left of the origin, and 0 verifying it has not moved vertically from the origin.

x-axis

The x-axis is a horizontal line on the coordinate plane that helps to establish a reference point for determining horizontal positions of points. It runs left-right across the plane and plays a crucial role in forming ordered pairs as the first component. By moving either to the left or right, the x-axis helps communicate whether the x-value increases or decreases:

• Moving right from the origin increases the x-coordinate.
• Moving left decreases the x-coordinate.

In our exercise, point D moves three units to the left on this axis. This means we subtract 3 from the origin, reflected in x = -3 .

y-axis

The y-axis is another primary reference line on the coordinate plane, but it runs vertically, unlike the x-axis. This axis indicates the vertical position of points and makes up the second part of the ordered pair. Here is a breakdown of movements along the y-axis:

• Moving upwards results in a positive increase in the y-coordinate.
• Moving downwards leads to a negative decrease.

In the context of point D, the y-coordinate remains 0, meaning there is no vertical movement from the origin. This stability confirms that the entire movement for D is horizontal.

origin

The origin is the central point of the coordinate plane, where the x-axis and y-axis intersect. It is represented by the ordered pair (0, 0) and acts as the starting point for locating other points within the plane. Key aspects of the origin include:

• It provides a baseline for measuring distance along both axes.
• All movements—left, right, up, or down—are calculated with respect to the origin.
• It offers a stable reference point, ensuring consistency for all coordinate-related work within the plane.

Understanding the origin's role is vital as it forms the basis for understanding how coordinates, like those of point D, change in relation to it. Point D adjusted its position leftward using the origin as a reference, indicating its final position through the ordered pair (-3, 0) .