Problem 10 - College Algebra Essentials

Question

In Exercises \(1-12\), plot the given point in a rectangular coordinate system. $$ (0,-3) $$

Step-by-Step Solution

Answer

The point (0,-3) is located on the y-axis, 3 units below the origin.

Step 1: Identify x and y coordinates

The first number in the point's coordinates represents the x coordinate and the second number is the y coordinate. In this case, we are given the point (0, -3). So, the x coordinate is 0 and the y coordinate is -3.

Step 2: Start at the origin

In the rectangular (or Cartesian) coordinate system, we always start plotting a point from the origin (0, 0). Because our x-coordinate is 0, we do not move left or right from the origin

Step 3: Move along the Y-axis

Since the y-coordinate is -3, move 3 units downward from the origin along the negative y-axis. The point (0,-3) is on the y-axis, 3 units below the origin.

Step 4: Plot the point

The point at which we end up is the location of our point (0,-3) in the rectangular coordinate system. Mark this point on the graph.

Key Concepts

Plotting Points

Plotting points on a rectangular coordinate system, also known as the Cartesian plane, is a way of visually representing numerical relationships. Each point is defined by an ordered pair of numbers, also called coordinates. Here, the objective is to place these coordinates accurately on the graph.

• Begin by understanding that a point is marked based on its location in a 2-dimensional space.
• The typical format of a point is \( (x, y) \), where 'x' is the horizontal component and 'y' is the vertical component.

The process of plotting a point involves starting at the origin, which is the center of the graph, and then moving according to the values of the x and y coordinates.
In exercises like plotting the point \( (0, -3) \), familiarity with how each coordinate adjusts your movement on the graph is crucial. As we interactively plot points, we improve our understanding of mathematical relationships and spatial awareness.

X and Y Coordinates

To effectively plot points, it is essential to comprehend what x and y coordinates are. Each provides a distinct piece of information:

• The x-coordinate indicates the movement along the horizontal axis.
• The y-coordinate indicates the movement along the vertical axis.

In the example point given, \( (0, -3) \), the x-coordinate is 0. This means there is no horizontal shift from the origin.
The y-coordinate is -3, instructing that you move down by three units from the zero point on the y-axis.
It's helpful to visualize these coordinates as directions or instructions—x gives you the left or right movement, while y provides the up or down shift. Together, they help you locate the exact position of a point on the graph.

Graphing on Cartesian Plane

Graphing on the Cartesian plane allows us to visualize points and derive meaningful insights from data. The Cartesian plane is a flat surface with two number lines at right angles.

• The x-axis runs horizontally, left to right.
• The y-axis runs vertically, up and down.

These axes intersect at the origin \( (0, 0) \), the central point of the graph.
Once we have our coordinates, we begin at the origin and plot based on the x and y values. Use the coordinate \( (0, -3) \) as an example.
On the Cartesian plane:

• Since the x-coordinate is 0, you remain on the y-axis.
• The y-coordinate of -3 instructs you to move three units downward.

This systematic movement from the origin allows you to accurately graph each point, facilitating a greater understanding of spatial relationships in mathematics.