Problem 34 - College Algebra with Corequisite Support

Question

Name the quadrant in which the following points would be located. If the point is on an axis, name the axis. (a) \((-3,-4)\) (b) \((-5,0)\) (c) \((1,-4)\) (d) \((-2,7)\) (e) \((0,-3)\)

Step-by-Step Solution

Answer

(a) Quadrant III, (b) x-axis, (c) Quadrant IV, (d) Quadrant II, (e) y-axis.

Step 1: Identify the Quadrants and the Axes

The Cartesian plane consists of four quadrants: Quadrant I (positive x, positive y), Quadrant II (negative x, positive y), Quadrant III (negative x, negative y), and Quadrant IV (positive x, negative y). The axes are the x-axis and y-axis.

Step 2: Determine the Location of Point (a)

The point (-3,-4) has negative x and negative y values. Points with both coordinates negative are located in Quadrant III.

Step 3: Determine the Location of Point (b)

The point (-5,0) has a negative x-value and a zero y-value. Points with zero y-values lie on the x-axis.

Step 4: Determine the Location of Point (c)

The point (1,-4) has a positive x value and a negative y value. Points with positive x and negative y are located in Quadrant IV.

Step 5: Determine the Location of Point (d)

The point (-2,7) has a negative x-value and a positive y-value. Points with negative x and positive y are located in Quadrant II.

Step 6: Determine the Location of Point (e)

The point (0,-3) has a zero x-value and a negative y-value. Points with zero x-values lie on the y-axis.

Key Concepts

Cartesian Coordinates

The Cartesian coordinate system allows us to describe the location of points in a two-dimensional plane. Each point on this plane is determined by a pair of numerical values, known as Cartesian coordinates. These coordinates take the form of an ordered pair \(x, y\), where \x\ represents the horizontal position along the x-axis, and \y\ signifies the vertical position along the y-axis.

The purpose of Cartesian coordinates is to give each point, line, or shape a precise location on the grid. In this system, positive x-values are located to the right of the origin, while negative x-values are to the left. Similarly, positive y-values rise above the origin, and negative y-values fall below. This grid system is utilized extensively in mathematics, physics, engineering, geospatial analysis, and various other fields, making it a fundamental element of mathematical comprehension.

Coordinate Axes

The coordinate axes form the basis of the Cartesian coordinate system, consisting of two perpendicular lines called the x-axis and y-axis. Together, these axes create four sections in the plane known as quadrants.

• X-Axis: The horizontal line that extends indefinitely to the left and right. It demarcates the plane's top from its bottom, handling all changes along the x-direction. When a point lies directly on the x-axis, its y-coordinate is zero.
• Y-Axis: The vertical line that stretches upwards and downwards without limit. It divides the plane into left and right regions, governing variations in the y-direction. Points that sit on the y-axis have an x-coordinate of zero.

The intersection point of these axes is called the origin, denoted by the coordinates \(0,0\). At the origin, both axes are zero, acting as the central point of reference that determines all other positions in the plane.

Identifying Quadrants

Identifying which quadrant a point belongs to in the Cartesian plane is straightforward once you understand the sign of its x and y coordinates. The plane is divided into four distinct quadrants that form around the origin:

• Quadrant I: Both x and y coordinates are positive (++++).
• Quadrant II: The x-coordinate is negative, while the y-coordinate is positive (-+++).
• Quadrant III: Both coordinates are negative (--).
• Quadrant IV: The x-coordinate is positive, but the y-coordinate is negative (++++).

If a point is directly on one of the axes, it doesn't belong to any quadrant. Instead, such a point is referred to simply by the axis it lies on. Understanding these relationships is crucial for mathematical navigation and problem-solving, as it determines the positioning of points concerning the origin.