Problem 11 - Algebra and Trigonometry

Question

In Exercises 1–12, plot the given point in a rectangular coordinate system. $$ \left(\frac{7}{2},-\frac{3}{2}\right) $$

Step-by-Step Solution

Answer

The point \((\frac{7}{2},-\frac{3}{2})\) is in the fourth quadrant and can be found by moving 3.5 units to the right on the x-axis and then 1.5 units down on the y-axis.

Step 1: Understand the Coordinate Notation

The coordinates of a point on a rectangular coordinate system are written in the format \((x,y)\), where \(x\) is the horizontal coordinate and \(y\) is the vertical coordinate. Looking at the given point, \((\frac{7}{2},-\frac{3}{2})\), it can be deduced that the point has an \(x\) coordinate of \(\frac{7}{2}\) and a \(y\) coordinate of \(-\frac{3}{2}\).

Step 2: Plotting the X-coordinate

Starting from the origin (0,0), move \(\frac{7}{2}\) or 3.5 units to the right direction. This is because \(\frac{7}{2}\) is a positive number which directs to the right on the \(x\)-axis.

Step 3: Plotting the Y-coordinate

From the point located on the \(x\)-axis, move \(-\frac{3}{2}\) or -1.5 units in the downward direction because \(-\frac{3}{2}\) is negative, which instructs to move down on the \(y\)-axis.

Step 4: Marking the point

After reaching the point, mark it on the graph to represent the given point \((\frac{7}{2},-\frac{3}{2})\). You will find that the point is located in the fourth quadrant because its \(y\)-coordinate is negative while its \(x\)-coordinate is positive.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a foundational concept in mathematics. It enables us to visually represent points and shapes in a two-dimensional plane. This system consists of two perpendicular number lines: the horizontal axis, called the x-axis, and the vertical axis, called the y-axis. These axes intersect at a point known as the origin, labeled as (0,0).

• The x-axis typically runs left to right, where values increase from the left towards the right.
• The y-axis traditionally runs up and down, where values increase from bottom to top.

In this system, any point can be described using an ordered pair denoted as (x, y), where 'x' specifies the point's position on the x-axis and 'y' indicates its position along the y-axis. These coordinates help us locate specific locations or plot graphs in a clear and standardized way.

Plotting Points

Plotting points in a rectangular coordinate system involves identifying the positions of 'x' and 'y' and marking them on the graph. The process starts by understanding the role of each part of the coordinate pair.

• The 'x' coordinate tells us how far and in which direction (left or right) to move from the origin.
• The 'y' coordinate indicates how far up or down to move from that new position on the x-axis.

For instance, plotting the point \( \left(\frac{7}{2}, -\frac{3}{2}\right) \), one starts at the origin. You would move \( \frac{7}{2} \) (or 3.5) units to the right, as it is positive, and then \(-\frac{3}{2}\) (or -1.5) units downward since it is negative. This precise movement ensures the point is correctly located on the grid.

Quadrants in Coordinate System

In any rectangular coordinate system, the plane is divided into four sections called quadrants. These quadrants provide a way to categorize points based on the signs of their coordinates. Starting from the top right and moving counterclockwise:

• **First Quadrant**: Both x and y are positive.
• **Second Quadrant**: x is negative, y is positive.
• **Third Quadrant**: Both x and y are negative.
• **Fourth Quadrant**: x is positive, y is negative.

Recognizing which quadrant a point belongs to helps in understanding its relative position in the coordinate plane. For example, the point \( \left(\frac{7}{2}, -\frac{3}{2}\right) \) is located in the fourth quadrant. This is because its x-coordinate is positive, while its y-coordinate is negative. Understanding quadrants helps immensely in graph analysis and solving geometric problems.