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Chapter 2: Problem 17

In which quadrant or on which axis is each point located? \((7,-2)\)

Short Answer

Expert verified
Fourth Quadrant

Step by step solution

01

Identify the Coordinates

Look at the given point \((7, -2)\). Identify the x-coordinate and the y-coordinate. Here, the x-coordinate is 7 and the y-coordinate is -2.
02

Determine the Sign of Each Coordinate

Check the signs of the x and y coordinates. For the point \((7, -2)\), the x-coordinate is positive (7 > 0) and the y-coordinate is negative (-2 < 0).
03

Locate the Quadrant

Use the signs of the coordinates to determine the quadrant: \(x > 0, y > 0\) 鈥� First Quadrant\(x < 0, y > 0\) 鈥� Second Quadrant\(x < 0, y < 0\) 鈥� Third Quadrant\(x > 0, y < 0\) 鈥� Fourth Quadrant. For \((7, -2)\), since the x-coordinate is positive and the y-coordinate is negative, the point is in the Fourth Quadrant.

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Key Concepts

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

Identifying Coordinates
Understanding coordinates is essential when working with the coordinate plane. Each point on the plane is defined by a pair of numbers, written as \(x,y\). These coordinates tell us where the point is located:
The first number, known as the x-coordinate, tells us how far to move left or right from the origin (the center point where the x and y-axes intersect).
The second number, the y-coordinate, tells us how far to move up or down from the origin.
For example, in the point \((7, -2)\), the x-coordinate is 7 and the y-coordinate is -2. This means you move 7 units to the right (since 7 is positive) and 2 units down (since -2 is negative) from the origin.
Sign of Coordinates
The sign of each coordinate is vital in determining the exact location of a point on the plane. Here鈥檚 what each sign tells us:
  • \((x > 0)\) - A positive x-coordinate means moving to the right from the origin.
  • \((y > 0)\) - A positive y-coordinate means moving up from the origin.
  • \((x < 0)\) - A negative x-coordinate means moving to the left from the origin.
  • \((y < 0)\) - A negative y-coordinate means moving down from the origin.

In the point \((7, -2)\), the x-coordinate is positive (7), which means moving right. The y-coordinate is negative (-2), which means moving down. These signs help in figuring out which quadrant or axis the point is located.
Locating Points
With the coordinate signs identified, locating the point is straightforward. The coordinate plane is divided into four quadrants:
  • First Quadrant: \(x > 0, y > 0\)
  • Second Quadrant: \(x < 0, y > 0\)
  • Third Quadrant: \(x < 0, y < 0\)
  • Fourth Quadrant: \(x > 0, y < 0\)

Points can also be on the axes:
- If \((x = 0)\), the point is on the y-axis.
- If \((y = 0)\), the point is on the x-axis.
For the point \((7, -2)\), since the x-coordinate is positive (right) and the y-coordinate is negative (down), it falls in the Fourth Quadrant.

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