Problem 34 - College Algebra

Question

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

Step-by-Step Solution

Answer

a. Third quadrant, b. Negative x-axis, c. Fourth quadrant, d. Second quadrant, e. Negative y-axis.

Step 1: Analyze Point (-3,-4)

For the point \((-3, -4)\), the x-coordinate is negative and the y-coordinate is also negative. This means the point is located in the third quadrant, since both coordinate values are negative in this quadrant.

Step 2: Analyze Point (-5,0)

Looking at the point \((-5, 0)\), the x-coordinate is negative and the y-coordinate is zero. Points that have a y-coordinate of zero lie on the x-axis. Hence, point \((-5,0)\) is on the negative x-axis.

Step 3: Analyze Point (1,-4)

The point \((1, -4)\) has a positive x-coordinate and a negative y-coordinate. This places it in the fourth quadrant, where the x-values are positive and the y-values are negative.

Step 4: Analyze Point (-2,7)

In this case, for point \((-2, 7)\), the x-coordinate is negative, and the y-coordinate is positive. This configuration indicates that the point is in the second quadrant, where x-values are negative and y-values are positive.

Step 5: Analyze Point (0,-3)

The point \((0, -3)\) has an x-coordinate of zero and a negative y-coordinate. Points with an x-coordinate of zero lie on the y-axis. Hence, this point is on the negative y-axis.

Key Concepts

Quadrants

In mathematics, the coordinate plane is divided into four sections, known as quadrants. These quadrants help us determine the position of a point based on its coordinates, which consist of an x-coordinate and a y-coordinate.

Each of the four quadrants has a unique combination of positive and negative signs for these coordinates:

• First Quadrant: Both x and y coordinates are positive.
• Second Quadrant: The x-coordinate is negative, while the y-coordinate is positive.
• Third Quadrant: Both x and y coordinates are negative.
• Fourth Quadrant: The x-coordinate is positive, while the y-coordinate is negative.

Understanding quadrants is crucial because it allows us to predict and verify the position of points on the coordinate plane.

For instance, the point \((-3, -4)\) is in the third quadrant where both coordinates are negative, and the point \((1, -4)\) is in the fourth quadrant where the x-coordinate is positive and the y-coordinate is negative.

Coordinate System

The coordinate system provides a way to specify the position of points on a plane. This system uses ordered pairs, typically written as \((x, y)\), where x indicates horizontal position and y indicates vertical position.

The coordinate system divides the plane into the four quadrants mentioned earlier, with lines intersecting at right angles to form the axes:

• The x-axis runs horizontally, serving as the reference for horizontal measurement.
• The y-axis runs vertically, serving as the reference for vertical measurement.

By using coordinates, you can precisely determine where a point lies. Points on this plane follow the pattern of reading from the x-coordinate to the y-coordinate.

The coordinate system not only aids in graphing but also helps in solving geometric problems and algebraic equations.

Axes Identification

Axes are essential parts of the coordinate plane as they serve as boundaries for the quadrants and assist in specifying pinpoint accuracy for points on the plane.

There are two main axes in the Cartesian coordinate plane:

• The x-axis: Identified as the horizontal line where the y-coordinate is zero. Points such as \((-5, 0)\) lie on this axis.
• The y-axis: Identified as the vertical line where the x-coordinate is zero. Points such as \((0, -3)\) lie on this axis.

When identifying a point on an axis:

• If the y-coordinate is zero, the point is on the x-axis.
• If the x-coordinate is zero, the point is on the y-axis.

For example, the point \((0, -3)\) lies on the negative side of the y-axis, while the point \((-5, 0)\) is located on the negative side of the x-axis.

Knowing how to identify and utilize these axes is essential for accurately plotting and understanding points on the coordinate plane.