Problem 11 - Prealgebra

Question

Graph each of the following ordered pairs. $$\left(5, \frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The point \((5, \frac{1}{2})\) is plotted by moving right 5 units and up \(\frac{1}{2}\) unit from the origin.

Step 1: Identify the Ordered Pair Coordinates

The ordered pair given is \((5, \frac{1}{2})\). The first number, 5, is the x-coordinate, and the second number, \(\frac{1}{2}\), is the y-coordinate. This means the point is located at 5 on the x-axis and \(\frac{1}{2}\) on the y-axis.

Step 2: Locate the x-coordinate on the Graph

To find the x-coordinate, start at the origin on the graph, where the x-axis and y-axis intersect. Move 5 units to the right along the x-axis, since the x-coordinate is positive.

Step 3: Locate the y-coordinate on the Graph

From the point located at 5 on the x-axis, move \(\frac{1}{2}\) units up, since \(\frac{1}{2}\) is positive on the y-axis. This will locate the y-coordinate.

Step 4: Plot the Point on the Graph

Once you've located both the x-coordinate and the y-coordinate, plot the point on the graph at \((5, \frac{1}{2})\). This is where the vertical line from the x-coordinate meets the horizontal line from the y-coordinate.

Key Concepts

Plotting Points

Plotting points on a graph is analogous to marking a specific location on a map. Each point on a graph is defined by an ordered pair of numbers, known as coordinates. In the ordered pair

• The first number is the x-coordinate
• The second is the y-coordinate

These coordinates help locate points precisely on the graph grid.
When plotting a point, you always begin by identifying these coordinates. Think of the x-coordinate as the number of steps you take to the right (or left for negative numbers) starting from the origin, which is the intersection of the axes. After that, you move vertically as indicated by the y-coordinate. Once correct movements are made, mark the dot where these two paths intersect.

X-axis and Y-axis

The x-axis and y-axis are the backbone of the coordinate system, forming a two-dimensional grid on every graph. The x-axis is the horizontal line that runs from left to right, while the y-axis is the vertical line that runs from bottom to top. These two axes intersect at a point known as the origin, designated as \( (0,0) \).
The task of plotting points involves moving parallel to these axes. When dealing with the x-coordinate, you measure rightward for positive numbers and leftward for negative numbers along the x-axis. In contrast, the y-coordinate requires movement upward for positive numbers and downward for negative numbers along the y-axis.
Thus, knowing how to navigate these axes is critical for accurately placing a point on the graph.

Coordinate System

A coordinate system is the framework that facilitates the precise plotting of points, lines, and curves on a plane. The most commonly used system is the Cartesian coordinate system, named after the mathematician René Descartes. This system consists of two intersecting perpendicular lines known as axes, creating four quadrants on a graph.

• The area where both coordinates are positive falls in the first quadrant.
• The remaining three quadrants accommodate negative values of one or both coordinates.

Each point has a unique coordinate pair, simplifying complex calculations and providing a visual layout within this system. By understanding how this grid operates, you can visualize numerical relationships and solve geometry problems more effectively.