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Chapter 10: Problem 87

Determine the quadrant in which the point is located without plotting it. $$(-1,-3)$$

Short Answer

Expert verified
The given point (-1,-3) is located in Quadrant III.

Step by step solution

01

Understand what each quadrant represents in terms of positive and negative values

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.
02

Analyze the given point

The point given is (-1,-3). Here, both x and y coordinates are negative.
03

Determine the quadrant

Since both coordinates of the given point are negative, this point is located in Quadrant III according to the information understood in Step 1.

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Key Concepts

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

Negative Coordinates
In the coordinate plane, negative coordinates are essential for locating points accurately. The coordinate plane consists of four quadrants, each defined by its combination of positive and negative values for the x-axis (horizontal) and y-axis (vertical).
  • A negative coordinate occurs when a value is to the left of the y-axis (for the x-coordinate) or below the x-axis (for the y-coordinate).
  • A point with negative x and y coordinates is located in the lower-left section of the coordinate plane.
Negative values are crucial as they broaden the coordinate plane beyond the positive-only limitations. This allows for a comprehensive mapping of geometric shapes and mathematical functions over the entire plane.
Understanding how negative coordinates compliment positive ones helps us see the full picture of spatial relationships.
Analyzing Points
Analyzing points involves examining coordinates to determine their position on the coordinate plane. Each point is represented by an ordered pair, which specifies its exact place based on the x (horizontal) and y (vertical) axes.
  • The first number in the pair is the x-coordinate. It indicates the horizontal distance from the origin (0,0). Negative values show a leftward movement while positive values indicate a rightward path.
  • The second number is the y-coordinate, guiding the vertical position. Negative numbers represent a movement downward from the origin, whereas positive numbers show an upward trajectory.
By thoroughly inspecting these coordinates, we gain insight into the position and context of each point in relation to the entire plane. This method is key when locating points, determining their orientation, and performing further calculations.
Quadrant Determination
Quadrant determination is the process of identifying which of the four sections, known as quadrants, holds a particular point based on its coordinates. The coordinate plane is divided into:
  • Quadrant I (top right): Both coordinates positive.
  • Quadrant II (top left): X is negative, Y positive.
  • Quadrant III (bottom left): Both coordinates negative.
  • Quadrant IV (bottom right): X is positive, Y negative.
By analyzing the signs of the coordinates, the position of the point can be quickly and accurately determined without needing to graphically plot them. For example, a point like (-1, -3) is immediately recognized to reside in Quadrant III, since both its x and y values are negative, according to the quadrant rules outlined.

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