Problem 12 - Thinking Mathematically

Question

Plot the given point in a rectangular coordinate system. \((0,-4)\)

Step-by-Step Solution

Answer

The point \((0,-4)\) is plotted 4 units below the origin along the y-axis since its x-coordinate is 0 and the y-coordinate is -4.

Step 1: Identify the coordinates

The point to plot is \((0,-4)\). In this pair, the first number (0) is the x-coordinate and the second number (-4) is the y-coordinate. Thus, the point we want to plot is on the y-axis at a distance of 4 units below the origin.

Step 2: Sketch the rectangular coordinate system

Start by sketching the x-axis (a horizontal line) and the y-axis (a vertical line) intersecting at the origin (the point \((0,0)\)). Label these axes for clarity.

Step 3: Plot the point

Starting from the origin (0,0), follow the y-axis in the negative direction since the y-coordinate of our point is negative. Because our y-coordinate is -4, we move 4 units below the origin and make a point to represent \((0, -4)\). Label the point.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a fundamental framework for plotting points, lines, and shapes on a flat surface. It consists of two perpendicular lines called axes, which intersect at a point known as the origin. On a typical 2-dimensional plane, we refer to these axes as the x-axis (horizontal) and y-axis (vertical).

Every point on this plane can be located using a pair of numerical coordinates: the x-coordinate tells us how far along the x-axis a point is, and the y-coordinate tells us how far along the y-axis. The origin is marked with the coordinates \(0,0\) and serves as the reference point from which all other points are measured.

Working with the rectangular coordinate system helps us visually represent mathematical concepts and analyze data in subjects like algebra, geometry, physics, and even economics.

X-axis and Y-axis

In the rectangular coordinate system, the x-axis and y-axis have distinct roles. The x-axis runs horizontally and is used to determine how far left or right a point lies from the origin. The y-axis, on the other hand, runs vertically, and it indicates how far up or down a point is from the origin.

These axes divide the plane into four quadrants, which are numbered counterclockwise starting from the upper right quadrant. Points can have positive or negative coordinates: a positive x-coordinate places a point to the right of the y-axis, while a positive y-coordinate places it above the x-axis. Conversely, negative x-coordinates are to the left of the y-axis, and negative y-coordinates are below the x-axis.

Graphing Coordinates

Graphing coordinates is a practical skill for visually interpreting data points on a rectangular coordinate system. To graph a coordinate, such as \(0,-4\), you'll first locate the x-coordinate on the x-axis. Since our x-coordinate is 0, this tells us the point is on the y-axis itself. Next, you will find the y-coordinate on the y-axis. The y-coordinate of \-4\ indicates the point is 4 units below the origin, because negative on the y-axis means moving downwards.

To represent this on a graph, you begin at the origin and move along the x-axis (horizontally) to the point that corresponds to the x-coordinate. Then, shift in the direction along the y-axis (vertically) as indicated by the y-coordinate and place a dot to mark the location. Always label the point after plotting to clearly indicate its position. Practice plotting points with different coordinates to become proficient at reading and creating graphs.