Problem 9 - Algebra and Trigonometry

Question

In Exercises 1–12, plot the given point in a rectangular coordinate system. $$ (-4,0) $$

Step-by-Step Solution

Answer

The point (-4,0) is located on the x-axis, four units to the left from the origin.

Step 1: Understand the Rectangular Coordinate System

In a rectangular coordinate system, we have two axis known as the x-axis (horizontal) and the y-axis (vertical). At their intersection, we have the origin which is represented as (0, 0). The position of a point in the coordinate system is given as an ordered pair (x, y). The first number, or x-coordinate tells you how far to move left or right from the origin, and the second number, or y-coordinate tells you how far to move up or down.

Step 2: Plot the Point

The given point is (-4, 0). This means mover 4 units to the left from the origin on the x-axis, because our x-coordinate is negative, and make no move on the y-axis, because our y-coordinate is zero. So we land on the point (-4, 0) on the graph.

Key Concepts

Coordinate Axes

The coordinate axes are the foundation of the rectangular coordinate system. This system comprises two lines that intersect at right angles, forming a grid that helps us locate points. These lines are called axes, consisting of the x-axis, which runs horizontally, and the y-axis, which runs vertically. The point where they meet is known as the origin, labeled as (0, 0). This is the reference point for all other coordinates on the plane.

Each axis divides the plane into four distinct quadrants, providing a framework for identifying the position of points in relation to the origin. The x-axis indicates the horizontal position of a point, while the y-axis indicates the vertical position. Thus, with these two axes, any point on the plane can be determined uniquely and precisely. This method of plotting is particularly useful in geometry and algebra for visualizing relationships between numbers and formulating equations.

X-coordinate

The x-coordinate is an essential component in the rectangular coordinate system, representing the horizontal position of a point. It is the first value in an ordered pair, like (-4, 0), and is positioned on the x-axis. This number indicates how far a point is from the origin, moving either to the left or right.

Let's break down how to interpret the x-coordinate:

• If the x-coordinate is positive, you move to the right from the origin.
• If the x-coordinate is negative, like in our example (-4, 0), you move to the left from the origin.
• If the x-coordinate is zero, the point lies directly on the y-axis.

The x-coordinate is crucial in shaping our understanding of a point's horizontal alignment on the grid and provides the first step in precisely locating where a point sits in space.

Y-coordinate

Next, we have the y-coordinate, the second element in an ordered pair, which indicates the vertical position of a point in the rectangular coordinate system. In the pair (-4, 0), the y-coordinate is 0.

Understanding the y-coordinate involves:

• Moving up from the origin if the y-coordinate is positive, showing how high a point is.
• Moving down if the y-coordinate is negative, showing how low a point is.
• If the y-coordinate is zero, the point lies exactly on the x-axis, as seen with the point (-4, 0).

Unlike the x-coordinate, which tells us about lateral movement, the y-coordinate focuses on vertical placement. When graphed, the y-coordinate helps complete the picture, offering a full view of a point's placement relative to both axes.

Plotting Points

Plotting points is the process of taking an ordered pair, like (-4, 0), and locating it on the coordinate plane. This activity helps translate abstract numerical information into a visual, easier-to-understand form.

Here's how to plot a point effectively:

• Start at the origin (0, 0).
• Use the x-coordinate to move horizontally. In our example, since x = -4, you would move 4 units to the left.
• Then, use the y-coordinate to move vertically. With y = 0, you make no vertical movement.

By following these steps, you accurately position the point (-4, 0) on the coordinate grid. Plotting points is a foundational skill in mathematics, aiding in graphing equations and visualizing data. Each plotted point tells a part of a larger story which can be decoded through systematic spatial reasoning.