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Chapter 5: Problem 6

For each of the following, graph the given points. $$ (4,0) $$

Short Answer

Expert verified
Plot the point (4,0) on the x-axis, 4 units to the right of the origin.

Step by step solution

01

Understand the Cartesian Coordinate System

The Cartesian coordinate system is made up of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Points are represented as \( (x, y) \), where \( x \) is the horizontal distance from the origin (0,0), and \( y \) is the vertical distance from the origin.
02

Identify the Coordinates

For the given point (4,0), \( x = 4 \) and \( y = 0 \). This tells us that the point is 4 units to the right of the origin along the x-axis and 0 units up or down the y-axis.
03

Plot the Point

On a graph, starting from the origin (0,0), move 4 units to the right along the x-axis. Since \( y = 0 \), there is no need to move up or down. Place a dot at the location where you end up, which represents the point (4,0).
04

Verify the Plot

Ensure the point is correctly placed at (4,0). This means that it is on the x-axis, precisely 4 units away from the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cartesian coordinate system
The Cartesian coordinate system is an essential concept in mathematics, particularly in graphing and geometry. It comprises two perpendicular lines called axes: the x-axis and the y-axis. These axes intersect at a point called the origin, denoted as (0,0). The x-axis runs horizontally, while the y-axis runs vertically.

Every point on this plane can be defined by an ordered pair \(x, y\), where \(x\) represents the horizontal position and \(y\) represents the vertical position.

This system allows us to pinpoint locations, trace functions, and solve geometric problems visually. Mastering this concept is integral to understanding more advanced topics in mathematics.
plotting points
Plotting points on a Cartesian coordinate system is both simple and foundational. To do this, you must follow a few straightforward steps:

Firstly, identify the coordinates of the point you need to plot. Coordinates are given in pairs \(x, y\), where \(x\) is the horizontal distance from the origin and \(y\) is the vertical distance.

In our example, for the point (4,0), \(x = 4\) and \(y = 0\). This means you'll move 4 units to the right on the x-axis, and since \(y = 0\), you won鈥檛 move up or down.

Next, starting from the origin (0,0), follow the coordinates to find the right spot on the grid. Place a dot at this precise location to represent the point.
x-axis and y-axis
The x-axis and y-axis are the backbone of the Cartesian coordinate system. The x-axis runs horizontally from left to right, and the y-axis runs vertically from bottom to top. Each axis is marked with numbers, which help us determine the positions of points.

To plot the point (4,0), begin at the origin where both axes intersect. Move 4 units to the right on the x-axis, as the x-coordinate is 4. Since the y-coordinate is 0, you don鈥檛 need to move vertically.

Thus, the point (4,0) lies on the x-axis, exactly 4 units away from the origin. Understanding the coordinate plane's structure will significantly ease your process of plotting and reading points.

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