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Chapter 4: Problem 87

In which quadrant is \(\left(-\frac{3}{2}, 15\right)\) located?

Short Answer

Expert verified
The point (-3/2,15) lies in the second quadrant.

Step by step solution

01

Analyze the given coordinates

The point given is (-3/2, 15). Here -3/2 is the x-coordinate and 15 is the y-coordinate. We need to recognize the sign of the coordinates, negative x-coordinate indicates left side of the origin, and positive y-coordinate indicates above the origin.
02

Determine the quadrant

Since the x-coordinate is negative and the y-coordinate is positive, the point lies in the quadrant where x is negative and y is positive. This quadrant is the second quadrant.

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Key Concepts

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

Understanding Quadrants
A coordinate system is divided into four sections known as quadrants, which appear like four equal parts of a plane. Each quadrant is identified by Roman numerals: I, II, III, and IV. Quadrants are counted in a counter-clockwise direction from the positive x-axis.

  • Quadrant I: Both x and y coordinates are positive.
  • Quadrant II: x is negative, y is positive.
  • Quadrant III: Both x and y coordinates are negative.
  • Quadrant IV: x is positive, y is negative.
Each quadrant represents a unique combination of positive and negative values for x and y, helping to precisely locate points in a plane.
Decoding the X-coordinate
The x-coordinate in a coordinate pair such as \((-\frac{3}{2}, 15)\), represents a point's horizontal position relative to the origin (0,0). It tells us how far left or right a point is located:

  • A positive x-coordinate indicates the point is to the right of the origin.
  • A negative x-coordinate signifies the point is to the left of the origin.
In this context, for the point \((-\frac{3}{2}, 15)\), the x-coordinate is \(-\frac{3}{2}\), positioning the point to the left on the coordinate system.
The Role of the Y-coordinate
The y-coordinate identifies a point's vertical position relative to the origin. It helps determine how far above or below the x-axis the point is located.

  • A positive y-coordinate means the point is above the origin.
  • A negative y-coordinate means the point is below the origin.
For the point \((-\frac{3}{2}, 15)\), the y-coordinate is \(15\), placing the point well above the x-axis.
Interpreting the Sign of Coordinates
The sign of the coordinates directly influences the quadrant in which a point is located. By examining the signs of a coordinate pair, we can determine the specific quadrant.
- A combination of positive x and positive y (positive, positive) places a point in Quadrant I.- A negative x with positive y results in Quadrant II.- Both negative x and y lead to Quadrant III.- Finally, positive x along with negative y indicates Quadrant IV.Consider the point \((-\frac{3}{2}, 15)\). Here, the negative x-coordinate and positive y-coordinate mean the point lies in the second quadrant, where the combination is found. This understanding helps in visualizing and predicting the point's location on a 2D plane.

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