Problem 8 - College Algebra Essentials

Question

In Exercises \(1-12\), plot the given point in a rectangular coordinate system. $$ (3,-2) $$

Step-by-Step Solution

Answer

The point (3,-2) is located three units to the right of the origin on the x-axis and two units down from that point on the y-axis.

Step 1: Understand the Coordinates

The exercise provides a pair of numbers (3,-2). In a rectangular coordinate system the first number represents the x-coordinate, and the second number is the y-coordinate. Therefore, the exercise asks to locate the point (3,-2) on the grid, which implies it is 3 units right from the origin on the x-axis and 2 units down on the y-axis.

Step 2: Draw the Rectangular Coordinate System

On a piece of paper, draw two perpendicular lines which intersect at a point known as the origin. Label the horizontal line as x-axis and the vertical line as y-axis. Include arrowheads on both ends of these lines to indicate that the lines extend indefinitely. Also split the axes into units to make plotting the point easier.

Step 3: Plot the Point on the Coordinate System

Plot a point three units to the right of the origin along the x-axis, that represents the x-coordinate. Then move two units down from this point (as the y-coordinate is negative), and mark this point. This is the required point, (3,-2).

Key Concepts

Plotting Points

Plotting points in a rectangular coordinate system is like playing a game of treasure hunt — you're following directions to find a precise location. Consider each point as part of a pair, known as coordinates, that help you locate a spot on a grid. This grid is made of an *x-axis*, which runs horizontally, and a *y-axis*, which runs vertically. Both axes intersect at a pivotal point called the *origin*. To plot a point, you use these two coordinates as directions.

• Locate the x-coordinate: Move along the x-axis the number of units indicated by the x-coordinate.
• Locate the y-coordinate: From the x-coordinate position, move vertically the number of units indicated by the y-coordinate.
• Mark the spot: This is where you plot your point.

Practice plotting various points to become fluent in navigating the rectangular coordinate system.

X-Coordinate

The x-coordinate is the first number in a pair of coordinates. This number tells you how far to move left or right from the origin along the x-axis. Imagine the x-axis as a "horizontal line". If the x-coordinate is positive, you move to the right from the origin. If it's negative, you move to the left. For example, if the x-coordinate is 3, you would start at the origin and count 3 units towards the right. This specific movement gives you a starting place to mark your y-coordinate.

• The x-coordinate is always the first number in the pair.
• It's crucial for determining the horizontal position of the point.

Being comfortable with identifying and interpreting the x-coordinate is essential to effectively plot points on the grid.

Y-Coordinate

The y-coordinate in a set of points tells you the position along the vertical line, which is your y-axis. Once you've moved based on your x-coordinate, the y-coordinate takes over, dictating how many units you need to move up or down from this new position. If the y-coordinate is positive, you move upwards; if negative, downwards. For instance, if the y-coordinate is -2, you would move 2 units down after aligning your x-coordinate. This action sets the exact spot for the point.

• The y-coordinate always follows the x-coordinate in the pair notation.
• Understanding how to interpret it helps accurately plot points vertically.

Mastering y-coordinate movements ensures precision when working with a coordinate plane.

Origin

The origin is the center of your coordinate system, where the x-axis and y-axis intersect. Located at (0,0), it acts as the starting point for any point plotting. Consider the origin like a very important landmark on this imaginary map. Understanding its role is fundamental when beginning to plot any points. Think of the origin as:

• The reference point from which all movements (left, right, up, down) are measured.
• The zero point for both the x-axis and y-axis, making it the anchor of the coordinate plane.

Familiarity with the origin helps demystify the plotting process and ensures you start every point placement on the right "foot." Given its significance, always ensure your interpretations of coordinates emanate from this crucial point.