Problem 12 - Prealgebra

Question

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

Step-by-Step Solution

Answer

Plot \(-5\) on x and move \(-\frac{1}{2}\) on y from there; mark the point.

Step 1: Understanding the Coordinate System

We start by identifying the components of the ordered pair, which in this case is \((-5, -\frac{1}{2})\). In any coordinate system, each point is represented as \((x, y)\). The first value \(-5\) represents the \(x\)-coordinate (horizontal position), while \(-\frac{1}{2}\) represents the \(y\)-coordinate (vertical position).

Step 2: Identifying the Coordinates

The \(x\)-coordinate is \(-5\), and the \(y\)-coordinate is \(-\frac{1}{2}\). The negative sign in the \(x\)-coordinate indicates it is positioned to the left of the y-axis, while the negative \(y\)-coordinate indicates it is below the x-axis.

Step 3: Plotting the x-Coordinate

To plot \(-5\), start at the origin (0,0) on your graph. Since \(-5\) is negative, move 5 units to the left along the x-axis.

Step 4: Plotting the y-Coordinate

From the point obtained in Step 3, move downwards (since \(-\frac{1}{2}\) is negative) by \(\frac{1}{2}\) unit along the y-axis.

Step 5: Marking the Ordered Pair

The point where the above movements intersect is the location of \((-5, -\frac{1}{2})\) on the coordinate plane. Place a dot or mark this point clearly, and label it if necessary to indicate it represents the given ordered pair.

Key Concepts

Understanding the Coordinate System

A coordinate system, like the Cartesian coordinate plane, is a method for identifying the exact position of points in a two-dimensional space. This system is made up of two number lines that intersect at a right angle, forming what we call an origin. The horizontal number line is called the x-axis and the vertical number line is called the y-axis. These axes divide the plane into four quadrants. Each point in this system is defined by an ordered pair \(x, y\), where \(x\) and \(y\) are the x-coordinate and y-coordinate, respectively.

Points along the x-axis will have a y-coordinate of zero, while points along the y-axis will have an x-coordinate of zero. Understanding this system is crucial, as it forms the basis for graphing and understanding ordered pairs.

What is the x-coordinate?

The x-coordinate is the first part of an ordered pair in a coordinate system. It indicates the horizontal position of a point relative to the origin.

Depending on the value, it tells us how far and in which direction we should move from the origin along the x-axis.

• Positive x-coordinates mean moving to the right of the origin.
• Negative x-coordinates indicate moving to the left of the origin.
• If the x-coordinate is zero, the point is exactly on the y-axis.

For example, in the ordered pair \(-5, -\frac{1}{2}\), the x-coordinate is \(-5\), directing us to move 5 units left from the origin.

Understanding the y-coordinate

The y-coordinate is the second part of an ordered pair, defining the vertical position of a point with respect to the origin. Moving vertically, this value directs us exactly where to plot along the y-axis.

• Positive y-coordinates mean moving above the x-axis.
• Negative y-coordinates indicate moving below the x-axis.
• An y-coordinate of zero implies the point lies on the x-axis.

In our given pair \(-5, -\frac{1}{2}\), the y-coordinate is \(-\frac{1}{2}\), which means we move downwards half a unit from the x-axis.

Plotting Points using Ordered Pairs

Plotting points is an essential skill in graphing ordered pairs on the coordinate plane. Here’s how you can do it:

Start with the origin point (0,0). First, use the x-coordinate. If it is negative or positive, move left or right respectively along the x-axis from the origin. Next, use the y-coordinate; move upwards for positive numbers and downwards for negative numbers once at the correct x-position.

Once you’ve moved along both axes, place a dot or mark at the location that both coordinates direct you to. This point represents the ordered pair.

Take, for example, the point \(-5, -\frac{1}{2}\): Begin at the origin, move 5 units left for the x-coordinate (-5), then move half a unit down for the y-coordinate (-\frac{1}{2}). Marking this spot correctly will give you the point \(-5, -\frac{1}{2}\) on the graph.