Problem 13 - Thinking Mathematically

Question

Plot the given point in a rectangular coordinate system. \(\left(-3,-1 \frac{1}{2}\right)\)

Step-by-Step Solution

Answer

The point \(-3, -1 \frac{1}{2}\) on a rectangular coordinate system would appear 3 units to the left of the origin on the x-axis and \(\frac{3}{2}\) units below the origin on the y-axis.

Step 1: Identify the coordinates

Identify the x and y coordinates from the given point \(-3, -1 \frac{1}{2}\). Here, the x-coordinate, which signifies horizontal movement, is -3, and the y-coordinate, which signifies vertical movement, is \(-1 \frac{1}{2}\). The negative x-coordinate means moving 3 units to the left from the origin (0,0) on the horizontal axis while the negative y-coordinate means moving \(\frac{3}{2}\) units downwards from the origin on the vertical axis.

Step 2: Plot the X-coordinate

The x-coordinate is -3. Starting from the origin (0,0), move 3 units to the left on the x-axis since the x-coordinate is negative.

Step 3: Plot the Y-coordinate

The y-coordinate is \(-1 \frac{1}{2}\). From the point on the x-axis (-3,0), move \(\frac{3}{2}\) units downward on the y-axis because the y-coordinate is negative.

Step 4: Mark the final point

Mark the final point at (-3,-1.5) representing the original given coordinates. This is the point of intersection after moving 3 units to the left from the origin on the x-axis and \(\frac{3}{2}\) units downward on the y-axis.

Key Concepts

Plotting Points

To plot points on a rectangular coordinate system, it's essential to comprehend both coordinates, which dictate the position of the point in a two-dimensional space. A point's position is expressed as an ordered pair

• (x, y), where the first value is the x-coordinate,
• the second is the y-coordinate.

In this context, the rectangular coordinate system is like a map that helps identify where a point is located based on how far it is from the origin. The origin is the central point

• commonly denoted as (0, 0),
• marking where the x-axis and y-axis intersect.

Each coordinate tells us how far the point is from these intersecting axes. To accurately plot a point, start at the origin, then move according to the coordinates.

X-Coordinate

The x-coordinate directs how far to move horizontally from the origin to plot a point. This value tells us whether to move left or right on the x-axis.
If the x-coordinate is negative, like -3, you move to the left.
If it is positive, you move to the right.

• Start at the origin (0, 0).
• Move horizontally along the x-axis.
• For an x-coordinate of -3, move three units left.

This horizontal movement places you directly on the x-axis at the new location in relation to its numerical value. This step sets the stage for placing a point in relation to the y-coordinate.

Y-Coordinate

Understanding the y-coordinate is crucial for determining vertical movement after positioning according to the x-coordinate. The y-coordinate indicates how far up or down to move.

• From your position on the x-axis,
• move depending on the y-coordinate's sign and value,
• since a negative y-coordinate like y = -1 \(\frac{1}{2}\)means moving downward.

In this case, starting from the location of (-3,0), descend 1.5 units vertically. This downward movement positions you exactly where the final point of (-3,-1 \(\frac{1}{2}\)) should be plotted on the graph.

Graphing Skills

Developing graphing skills involves repeated practice and familiarizing yourself with moving along the axes based on given coordinates. Here are some tips:

• Always start at the origin when plotting new points.
• Visualize the movement required by each coordinate.
• For negative values, remember it indicates moving in the opposite direction (left for x, down for y).
• Use a ruler to ensure movements are precise and neat on paper.

Continual practice plotting various points enhances understanding of geometric relationships and spatial awareness. Mastering these skills allows for accurate graphing and a strong foundation in mathematical concepts.