Problem 12 - College Algebra

Question

Plot the given point in a rectangular coordinate system. $$ \left(-\frac{5}{2}, \frac{3}{2}\right) $$

Step-by-Step Solution

Answer

The given point (-5/2, 3/2) is plotted by starting at the origin, moving -5/2 units to the left along the x-axis, and then from that point moving 3/2 units up along the y-axis.

Step 1: Understanding the coordinates

In an ordered pair \((x, y)\), the first number \(x\) represents the distance along the x-axis (horizontal axis) and the second number \(y\) represents the distance along the y-axis (vertical axis). If the number is negative, move in the opposite direction. Here, -5/2 is negative, which means moving to the left side of the x-axis and 3/2 is positive, which means moving to the up on y-axis.

Step 2: Plotting the point

Start from the origin (where the x and y axes intersect), then move -5/2 units to the left along the x-axis, and then from that point move 3/2 units up along the y-axis. Mark this point.

Step 3: Verification

Verify that the point is at the right location. The point's x-coordinate should be -5/2 units from the origin along the x-axis and the point's y-coordinate should be 3/2 units from the origin along the y-axis. If these conditions are met, then the point has been plotted correctly on the graph.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to locate points on a plane using two numbers. This system consists of two perpendicular lines called axes: the horizontal line is the x-axis, and the vertical line is the y-axis. The point where these lines intersect is called the origin, represented by \(0,0\). Rectangular coordinates allow us to describe a point’s position by how far it is along the x-axis (left or right) and the y-axis (up or down). This is an essential tool in mathematics and various fields like physics, engineering, and computer graphics.

Plotting Points

Plotting points on a rectangular coordinate system involves simple steps. First, you identify the x-coordinate and the y-coordinate from the ordered pair. For the point \(-\frac{5}{2}, \frac{3}{2}\), you start at the origin. Move \(-\frac{5}{2}\) units left along the x-axis. Then, move \(+\frac{3}{2}\) units up along the y-axis. Where you land after these movements is your point. Always double-check to ensure accuracy, confirming that each coordinate direction has been followed precisely.

X-axis and Y-axis

The x-axis is the horizontal line that runs left and right. The y-axis is the vertical line that runs up and down. Together, they form the coordinate system used for plotting points. Each axis allows you to measure distance in positive and negative directions. For example, negative values on the x-axis move to the left from the origin, whereas positive values move to the right. For the y-axis, negative values move down, and positive values move up. Understanding these directions helps plot any point confidently.

Ordered Pair

An ordered pair \(x, y\) represents a point’s exact location in the rectangular coordinate system. The first value, \(x\), indicates the position along the x-axis, while the second value, \(y\), indicates the position along the y-axis. In the ordered pair \(-\frac{5}{2}, \frac{3}{2}\), \(-\frac{5}{2}\) is the x-coordinate, and \(\frac{3}{2}\) is the y-coordinate. The sequence of these coordinates matters because it specifies direction: first along the x-axis, then along the y-axis. Understanding ordered pairs is crucial for graphing points accurately.