Problem 12 - Algebra and Trigonometry

Question

In Exercises 1–12, plot the given point in a rectangular coordinate system. $$ \left(-\frac{5}{2}, \frac{3}{2}\right) $$

Step-by-Step Solution

Answer

The point can be plotted in rectangular coordinate system by moving 2.5 units to the left of the origin for the x-coordinate and 1.5 units upward for the y-coordinate.

Step 1: Identify the coordinates

The point provided in the exercise is (-5/2, 3/2). Here, -5/2 is the x-coordinate and 3/2 is the y-coordinate.

Step 2: Plot the x-coordinate

Start at the origin, which is the point (0,0). Since the x-coordinate is -5/2, which is -2.5, move 2.5 units to the left of the origin.

Step 3: Plot the y-coordinate

From the new position, since the y-coordinate is 3/2, or 1.5, move 1.5 units upward.

Step 4: Mark the point

Mark the point at this location, which is (-5/2, 3/2), in the coordinate system.

Step 5: Check the point

Verify the plotted point corresponds to the coordinates given in the problem, ensuring the point is plotted correctly.

Key Concepts

Plotting Points

Plotting points is an essential skill in understanding how data and relationships are visualized in mathematics. When you plot a point, you are positioning a marker on a graph that corresponds to a given set of coordinates. Every point in a rectangular coordinate system is defined with an ordered pair.

• The first element is the x-coordinate, which indicates how far along the horizontal axis (left-right direction) the point is.
• The second element is the y-coordinate, representing how far along the vertical axis (up-down direction) the point resides.

The origin, or the center of the coordinate system, is at (0,0). This is your starting point for plotting. To plot the point (-5/2, 3/2), you start at the origin and move according to the values of the coordinates. Careful plotting helps in accurate graph representation and error-free interpretation.

X-Coordinate

The x-coordinate is a critical component of plotting on the coordinate grid because it determines the horizontal position of a point. It is the first number in the ordered pair. In the point (-5/2, 3/2), -5/2 is the x-coordinate. To plot the x-coordinate:

• Begin at the origin (0,0).
• Move left if the x-coordinate is negative, as seen here with -5/2 (meaning 2.5 units to the left).
• Move right if the x-coordinate is positive.

This step sets the foundation for where the point will vertically adjust based on the y-coordinate.

Y-Coordinate

After positioning using the x-coordinate, the next step is guided by the y-coordinate, which determines the vertical location of the point. The y-coordinate in our example is 3/2, meaning 1.5. To plot the y-coordinate:

• From the x-coordinate position, move upward if the y-coordinate is positive, just like moving 1.5 units up for 3/2.
• Move downward if the y-coordinate is negative.

By combining the movements dictated by both coordinates, you effectively mark the point's location on the graph.

Graphing Points

Graphing points involves integrating the plotting process to visually see where points fall within a grid-based system. It's like placing a dot at the precise intersection of where the x and y adjustments meet. Once you have both the x and y movements calculated, mark the exact spot. This spot corresponds to the specific pair of coordinates given in the exercise. Steps to ensure accuracy when graphing:

• Double-check both the x and y movements to ensure they are in the correct direction.
• Verify against the coordinate labels on the axes.
• Clearly mark and label the point (e.g., (-5/2, 3/2)) to avoid confusion in further analysis.

When points are graphed correctly, they provide a powerful visual tool for analyzing trends, relationships, and patterns in mathematical data.