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Chapter 7: Problem 113

In which quadrant is the point \(\left(6,-\frac{1}{2}\right)\) located?

Short Answer

Expert verified
The point (6, -1/2) is in the fourth quadrant.

Step by step solution

01

- Identify the Coordinates

The point given is \(\textbf{(6,-\frac{1}{2})}\). Here, 6 is the x-coordinate and \-\(\frac{1}{2}\) is the y-coordinate.
02

- Determine the Sign of the Coordinates

Check the signs of the x and y coordinates. The x-coordinate \(\textbf{(6)}\) is positive and the y-coordinate \(-\frac{1}{2}\) is negative.
03

- Identify the Quadrant Based on the Signs

Based on the signs, a point with a positive x-coordinate and a negative y-coordinate is located 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
Coordinates are a set of values that show an exact position on a plane. In a 2D coordinate system, each point is represented as \((x, y)\). The first value is the x-coordinate, which tells us how far to move left or right. The second value is the y-coordinate, indicating how far to move up or down.

For example, the point \((6, -\frac{1}{2})\) has an x-coordinate of 6 and a y-coordinate of -\frac{1}{2}. To identify coordinates:
  • Look at the ordered pair. The first number is the x-coordinate.
  • The second number is the y-coordinate.
This helps in understanding the arrangement and position of points in the coordinate plane.
Signs of Coordinates
The signs of the coordinates (positive or negative) tell us which direction to move. Here's a quick guide:
  • Positive x-coordinate means moving right.
  • Negative x-coordinate means moving left.
  • Positive y-coordinate means moving up.
  • Negative y-coordinate means moving down.
In the given point \((6, -\frac{1}{2})\), the x-coordinate is positive (6), so we move right. The y-coordinate is negative (-\frac{1}{2}), so we move down.

Knowing the signs helps in determining the exact location of the point in one of the four quadrants.
Quadrant Determination
The coordinate plane is divided into four quadrants:
  1. First Quadrant (I): Both x and y coordinates are positive.
  2. Second Quadrant (II): x is negative, y is positive.
  3. Third Quadrant (III): Both x and y coordinates are negative.
  4. Fourth Quadrant (IV): x is positive, y is negative.
To determine the quadrant of a point, check the signs of its coordinates. For the point \((6, -\frac{1}{2})\):
  • x is positive (6) -> Right direction
  • y is negative (-\frac{1}{2}) -> Down direction
Hence, this point is in the Fourth Quadrant (IV), as the x-coordinate is positive and the y-coordinate is negative.

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