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Chapter 2: Problem 30

Graph and label each point on a coordinate plane. Name the quadrant in which each point is located. $$C(6,-1)$$

Short Answer

Expert verified
The point \(C(6, -1)\) is in Quadrant IV.

Step by step solution

01

Understand Quadrants

The coordinate plane is divided into four quadrants. Quadrant I is where both x and y coordinates are positive. Quadrant II has a negative x and a positive y. Quadrant III has both coordinates negative. Quadrant IV has a positive x and a negative y. Our goal is to identify in which quadrant the point \(C(6, -1)\) is located.
02

Plot the Point

To plot the point \(C(6, -1)\) on the coordinate plane, locate the x-coordinate 6 on the horizontal axis (to the right of the origin), and the y-coordinate -1 on the vertical axis (below the origin). Mark this point on the plane.
03

Determine the Quadrant

For point \(C(6, -1)\), the x-coordinate is positive, and the y-coordinate is negative. This places point \(C\) in Quadrant IV, where all points have a positive x and a negative y.

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Key Concepts

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

Graphing Points
When you are graphing points on a coordinate plane, it is like playing a game of treasure hunt where the point is your treasure. Every point is determined by two numbers called coordinates. These coordinates are written like this: \(x, y\). The first number is the x-coordinate, which tells you how far to move left or right from the point called the origin, which is at \(0,0\). The second number is the y-coordinate, which tells you how far to move up or down from the origin. Let's break down this process in simple steps:
  • Start at the origin point \(0, 0\), where the x-axis and y-axis intersect.
  • Look at the x-coordinate of your point. If it is positive, move to the right. If it is negative, move to the left.
  • Next, look at the y-coordinate. If it is positive, move up. If it is negative, move down.
  • After you have moved according to both the x and y values, mark your point by drawing a dot and label it with its name, like \(C(6, -1)\).
This will help you visualize exactly where your point lies on the plane!
Quadrants
A coordinate plane is divided into four sections called quadrants. Imagine a cross that divides the plane into four pieces. Each of these four sections is called a quadrant, and they are numbered using Roman numerals: I, II, III, and IV. Here's how each quadrant is categorized:
  • Quadrant I: The x and y values are both positive. Points in this quadrant reside in the upper right part of the plane.
  • Quadrant II: The x-coordinate is negative while the y-coordinate is positive. This implies that points here are in the upper left section.
  • Quadrant III: Both coordinate values are negative, placing these points in the lower left quadrant.
  • Quadrant IV: The x-coordinate is positive, and the y-coordinate is negative. The points fall in the lower right section.
Understanding quadrants is key to knowing where your point resides. For example, in the point \(C(6, -1)\), it is in Quadrant IV because the x is positive and the y is negative. It's like having your own map to locate treasures in each section of your chart!
Cartesian Coordinates
Cartesian coordinates allow us to precisely locate any point on the coordinate plane using a specific ordered pair: \(x, y\). This system is named after the French mathematician Ren茅 Descartes, who pioneered this concept. Here's how it functions:
  • Each point on the plane is uniquely identified by these two numbers.
  • The x-value tells us how far left or right the point is from the origin: positive values move right, negative values move left.
  • The y-value indicates how far up or down the point is from the origin: positive values go up, and negative values go down.
Using Cartesian coordinates makes graphing an intuitive process and helps us to plot points methodically. For example, the point \(C(6, -1)\) means you start at the origin, move 6 steps to the right on the x-axis, and then 1 step down on the y-axis. This structured system allows us to easily and efficiently find any point in the coordinate plane.

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