Problem 13 - Algebra for College Students

Question

Plot the following points in a rectangular coordinate system. For each point, name the quadrant in which it lies or the axis on which it lies. $$(\pi,-1)$$

Step-by-Step Solution

Answer

The point \( \pi, -1\) lies in the Fourth Quadrant.

Step 1 - Understand the Coordinate System

A rectangular coordinate system consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Points are plotted as \((x, y)\), where \x\ is the position on the x-axis and \y\ is the position on the y-axis.

Step 2 - Plot the Point \( \pi, -1\)

To plot \( \pi, -1\) follow these instructions:1. Locate \pi\ on the x-axis. Since \pi\ is approximately 3.14, estimate the point slightly past 3 on the x-axis.2. From \pi\ on the x-axis, move vertically downward (since -1 is negative) to reach -1 on the y-axis.3. Mark the point where these two positions meet.

Step 3 - Determine the Quadrant or Axis

Review the location of the point \( \pi, -1\):- Since \pi\ (x-coordinate) is positive and -1 (y-coordinate) is negative, the point lies in the Fourth Quadrant.

Key Concepts

plotting points

Plotting points in a rectangular coordinate system is a fundamental skill in mathematics. In this system, each point is identified by an ordered pair, \( (x, y) \), which tells you where the point is located on the grid. The first number in the pair is the x-coordinate, which indicates the position along the horizontal axis. The second number is the y-coordinate, which shows the position along the vertical axis.

To plot a point, you follow these steps:

• Start at the origin, where the x-axis and y-axis intersect (0,0).
• Move horizontally to the x-coordinate.
• From that position, move vertically to the y-coordinate.
• Finally, mark the spot where this line meets the y-coordinate.

For example, to plot the point \( (\backslashpi, -1) \), find \( \backslashpi \) (approximately 3.14) on the x-axis, then move down to -1 on the y-axis and mark the point. This systematic approach ensures accuracy and clarity.

coordinate axes

The coordinate axes are the two lines that form the basis of the rectangular coordinate system. These are the x-axis and the y-axis. These axes are perpendicular to each other and intersect at a point called the origin, labeled (0,0). Understanding the axes is crucial for navigating the coordinate system effectively.

- The x-axis is the horizontal line that runs left to right.
- The y-axis is the vertical line that runs up and down.

Each axis is divided into positive and negative directions:

• The right side of the x-axis (positive direction).
• The left side of the x-axis (negative direction).
• The top half of the y-axis (positive direction).
• The bottom half of the y-axis (negative direction).

This division helps in correctly identifying the position of any point and its respective coordinates. It forms the structure on which all points are plotted.

quadrants

Quadrants divide the coordinate plane into four sections based on the signs of the coordinates. These quadrants are labeled counterclockwise starting from the upper right:

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

This division helps in quickly identifying the location of points. For example, the point \( \backslashpi, -1 \) is located in the Fourth Quadrant because \( \backslashpi \) is positive and -1 is negative. Understanding these quadrants is essential for graphing and interpreting points accurately.