Problem 27 - College Algebra with Corequisite Support

Question

For each of the following exercises, identify the information requested. What are the coordinates of the origin?

Step-by-Step Solution

Answer

The coordinates of the origin are (0, 0).

Step 1: Understanding the Task

The task asks us to identify the coordinates of the 'origin'. In a Cartesian coordinate system, the origin is the point where the horizontal x-axis and the vertical y-axis intersect.

Step 2: Identifying the Coordinates

In mathematics, the origin is at the point where both the x and y coordinates are zero. This means that the coordinates of the origin are (0, 0).

Key Concepts

Coordinates

In the Cartesian coordinate system, coordinates are used to describe the location of a point in a two-dimensional plane. They are always presented as an ordered pair, typically written as \((x, y)\). The first number, \(x\), represents the position of the point along the horizontal x-axis. The second number, \(y\), indicates the position along the vertical y-axis.
This pair of numbers helps us pinpoint exact locations on the grid, much like street addresses help us find a house in a city. Coordinates can be positive or negative, which tells us in which quadrant of the plane the point is located.

• Positive \(x\) and \(y\) place the point in Quadrant I.
• Negative \(x\) and positive \(y\) place the point in Quadrant II.
• Negative \(x\) and \(y\) place the point in Quadrant III.
• Positive \(x\) and negative \(y\) place the point in Quadrant IV.

Origin

The origin is a fundamental concept in the Cartesian coordinate system. It is the central point where the x-axis and y-axis intersect. At the origin, the values for both axes are zero, making its coordinates \((0, 0)\).This point acts as a reference from which all other points on the plane can be determined.
Understanding the origin is crucial because it serves as the benchmark for locating other points on the grid. When you start measuring a location, your starting point is always the origin. Here are some important traits of the origin:

• It is the middle point in the coordinate plane.
• All calculations of distance from other points can be measured beginning at the origin.
• It helps in defining symmetry of shapes and graphs.

X-Axis

The x-axis is the horizontal line in the Cartesian coordinate system. It's a critical part of plotting points and reading their coordinates.

The x-axis serves as the baseline for measuring horizontal distances. When you look at a coordinate pair, the first number corresponds to the location on this axis. For instance, in the coordinate \((3, 2)\), the number \(3\) tells you how far to move from the origin along the x-axis.

• Positive values move to the right of the origin.
• Negative values move to the left of the origin.
• The axis itself is typically represented by \(x = 0\), where only the y-values change.

The x-axis, alongside the y-axis, divides the plane into four quadrants and helps you figure out a point's location quickly.

Y-Axis

Perpendicular to the x-axis, the y-axis is the vertical line in the Cartesian coordinate system. The y-axis is used for measuring vertical distances relative to the origin.
Each coordinate \((x, y)\) uses the second number, \(y\), to specify movement along this axis. In the pair \((3, 2)\), \(2\) indicates the position above or below the x-axis.
The y-axis has some key characteristics:

• Positive y-values are above the x-axis.
• Negative y-values extend below the x-axis.
• Like the x-axis, it passes through the origin, making it \(y = 0\) when only x-values change.

Together with the x-axis, the y-axis forms the essential framework for graphing and analyzing points in the Cartesian system.