Problem 10 - Thinking Mathematically

Question

Plot the given point in a rectangular coordinate system. \((-5,0)\)

Step-by-Step Solution

Answer

The plotted point (-5,0) lies on the x-axis, 5 units to the left of the origin.

Step 1: Identify the Given Point

The given point is (-5,0). This means it is 5 units to the left of the origin on the x-axis (since x is -5) and it lies on the x-axis itself (since y is 0).

Step 2: Draw the rectangular coordinate system

Draw two perpendicular lines which intersect at a point. This point is called the origin. The horizontal line is the x-axis and the vertical line is the y-axis.

Step 3: Plot the given point

Starting at the origin, move 5 units to the left along the x-axis as the x-coordinate is -5. Since the y-coordinate is 0, there is no need to move up or down. Mark the position (-5,0) on the x-axis.

Key Concepts

plotting points

Plotting a point on a graph involves locating that point on the coordinate grid based on its given coordinates. To do this effectively, you need to understand the basic structure and terminology of the coordinate system.
The coordinates of a point are typically expressed in the form \(x, y\), with \(x\) representing the horizontal position relative to the origin and \(y\) representing the vertical position. When you have a point with these coordinates, follow the steps below to plot it accurately:

• Start at the origin, which is where the x-axis and y-axis intersect.
• Move horizontally by the value of the x-coordinate; move to the right if it's positive and to the left if it's negative.
• Then, move vertically by the value of the y-coordinate; move up if it's positive and down if it's negative.
• Mark the point where your movements end on the grid.

The accuracy in plotting these points is crucial as it serves as the foundation for more complex graphing tasks.

rectangular coordinate system

The rectangular coordinate system, also known as the Cartesian plane, is a two-dimensional grid used to locate points. It consists of two lines intersecting at right angles:

• The horizontal line is called the x-axis.
• The vertical line is called the y-axis.

These lines divide the plane into four quadrants. The intersection of the x-axis and y-axis is the origin, usually denoted by \( (0,0) \). This system allows us to specify a point’s exact location using a pair of numbers.

The rectangular coordinate system is widely used in mathematics due to its simplicity and precision. It can easily represent mathematical functions, physical locations, and more. By understanding this system, we can better interpret and visualize mathematical data and functions.

x-axis and y-axis

The x-axis and y-axis serve as the fundamental components of the rectangular coordinate system. These axes are basically number lines that intersect each other at a point called the origin. Here’s a closer look at each:

• **x-axis**: This is the horizontal axis in the system. It extends infinitely to the left and right. Points on the x-axis have a y-coordinate of zero, indicating no vertical movement from the origin.
• **y-axis**: The y-axis is the vertical counterpart, extending infinitely upwards and downwards. Points on the y-axis have an x-coordinate of zero.

The positive x direction is to the right of the origin, while the negative x direction is to the left. Similarly, the positive y direction is above the origin, and the negative direction is below. Understanding these directions and how to navigate them is crucial for successfully plotting points and interpreting graphs.