Problem 10 - College Algebra

Question

Plot the given point in a rectangular coordinate system. $$ (0,-3) $$

Step-by-Step Solution

Answer

The point (0,-3) is located 3 units below the origin in the rectangular coordinate system.

Step 1: Understand the Coordinates

The point is given as (0,-3). In a coordinate system, the coordinates are given in the order (x,y). So, in this case, x=0 and y=-3.

Step 2: Plot the x-coordinate

Plot x=0 on the x-axis. Note that x=0 is the point where the x-axis intersects the y-axis, known as the origin.

Step 3: Plot the y-coordinate

To plot the y-coordinate, move downwards from the origin (since y value is negative) by 3 units. This point, located on the vertical line x=0, is the position of the point (0,-3) in a rectangular coordinate system.

Key Concepts

Coordinate System

When faced with the task of plotting points, it's essential to understand what a coordinate system is. To put it simply, a coordinate system is like a map for mathematics that allows us to pinpoint the exact location of a point in space. Most commonly, we use what's called a Cartesian coordinate system, which is made up of two axes perpendicular to each other: the horizontal x-axis and the vertical y-axis.

The intersection where the two axes meet is called the origin, marked as (0,0). Every point in this system is represented by a pair of numbers, known as coordinates. The x-coordinate tells us how far along the horizontal axis the point is, while the y-coordinate tells us how far along the vertical axis the point moves. This system is incredibly useful not only in math but also in various fields like physics, engineering, and even computer graphics.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most fundamental way we communicate position in a two-dimensional plane. They allow us to specify locations using two numbers representing horizontal and vertical positions.

These coordinates are written in the form \( (x, y) \), where \( x \) and \( y \) correspond to positions on the x-axis and y-axis, respectively. When \( x \) is zero, the point lies directly on the y-axis, and when \( y \) is zero, the point is located on the x-axis. Understanding how these two values relate to the position on the grid is crucial for graphing. For example, the \( (0,-3) \) you're working with means that the point is at the origin horizontally \( (x = 0) \) and three units down vertically, as indicated by the negative sign before the 3.

Graphing Points

The process of graphing points on a coordinate system is straightforward once you understand the coordinates. To graph a point, you start by drawing a coordinate system with a horizontal x-axis and a vertical y-axis. Once you have the point's coordinates, like \( (0,-3) \), you begin at the origin and follow the steps according to the coordinates.

For the x-coordinate, if it's positive, you'd move to the right. If it's negative, you'd move to the left. With a zero x-coordinate, as in our example, you'd stay at the origin along the horizontal axis. Then, you move vertically according to the y-coordinate: up for positive and down for negative. For our point \( (0,-3) \), you would move three units downwards from the origin. Plotting this point accurately is crucial as it is the foundation for more complex operations like graphing lines and curves. If you're using graph paper, each unit can correspond to one square on the paper, which helps in maintaining precision.

Remember, treating these tasks like a treasure hunt—where 'X marks the spot'—can add an element of fun to the learning process and help reinforce these important graphing skills.