Problem 7 - College Algebra Essentials

Question

In Exercises \(1-12\), plot the given point in a rectangular coordinate system. $$ (4,-1) $$

Step-by-Step Solution

Answer

For the point (4,-1), move 4 units to the right along the x-axis from the origin, then 1 unit downward along the y-axis.

Step 1: Understand the Rectangular Coordinate System

In a rectangular coordinate system, every point is specified by a pair of numerical coordinates. These are the distances of the point from two perpendicular directed lines, called axes. We have 'x-axis' which is horizontal and 'y-axis' which is vertical. To the right and above the point where the axes intersect are positive directions, to the left and below are negative directions.

Step 2: Plot the Abscissa of the Point

The abscissa (x value) of the point (4,-1) is 4. Start at the origin (the point (0,0)) and move 4 units to the right along the x-axis.

Step 3: Plot the Ordinate of the Point

The ordinate (y value) of the point (4,-1) is -1. Starting from the point you reached in step 2, move 1 unit down along the y-axis, in negative direction. This will be the final point.

Step 4: Marking the Point

This point, which is four units to the right and one unit down from the origin on the coordinate plane, represent the coordinates (4,-1). Mark this point on the coordinate system.

Key Concepts

Rectangular Coordinate System

Imagine drawing a map to find a treasure in your backyard. You'd need a system to specify exact locations. Similarly, in mathematics, we use the rectangular coordinate system as a map to find points on a graph. This system comprises two perpendicular lines called axes, dividing the plane into four sections known as quadrants.

These axes intersect at a central point called the origin, with coordinates (0,0). The horizontal axis is known as the x-axis, and the vertical one is called the y-axis. The position of any point on this grid is determined by two numbers, also known as the point's coordinates. These coordinates tell us how far we need to move from the origin along the x-axis and then along the y-axis to reach our desired 'treasure'—the point in question.

To remember which way is positive, think

• Right along the x-axis is positive.
• Up along the y-axis is positive.

Conversely, going left on the x-axis or down on the y-axis will land you in the negative territory. This creates a unique and orderly system where every point can be precisely located and plotted.

Abscissa and Ordinate

Every treasure map has coordinates like '10 steps north, 5 steps east'. In our graph, these steps are called the abscissa and the ordinate, respectively. The abscissa refers to the x-coordinate—the number of units you move horizontally from the origin. Think of it as walking east or west from your starting point.

If you have a point (4, -1), the number 4 is the abscissa. To find this on the graph, you'd start at the origin and take 4 steps to the right because it's positive.

Next comes the ordinate, representing the y-coordinate—the vertical journey from the origin, much like climbing a ladder or descending into a cave. In our example, -1 is the ordinate. Since it's negative, you'd move 1 unit down from where you are on the x-axis. By combining these two movements, right from the origin and then down, you'd arrive exactly where you need to be on the graph—to your treasure point.

Coordinate Plane Graphing

Now, with our coordinate map and steps in mind, let's go on a quest to plot points. The process of coordinate plane graphing involves drawing a point on the graph that corresponds to its coordinates. To successfully embark on this adventure, follow these steps:

Step 1: Find the Abscissa

Identify the x-coordinate (abscissa) of your point. If it's positive, start at the origin and move to the right; if it's negative, move to the left. For the point (4, -1), we would move 4 units to the right.

Step 2: Locate the Ordinate

Now, from your position along the x-axis, look at the y-coordinate (ordinate). If it's positive, move up; if negative, move down. From where we stopped at the abscissa, since our ordinate is -1, we go down one unit.

Step 3: Plot and Mark the Point

Where these two steps intersect, place a dot. That's your point! In our case, put a dot to mark the point (4, -1). It's as simple as that. You have now charted your point on the treasure map of the coordinate plane. Always remember to double-check your steps to ensure your plotted point matches its coordinates.