Problem 62 - Precalculus Essentials

Question

Explain how to plot a point in the rectangular coordinate system. Give an example with your explanation.

Step-by-Step Solution

Answer

A point in the rectangular or Cartesian coordinate system is represented by an ordered pair \((x, y)\). You plot the point by starting at the origin (0,0), moving 'x' units horizontally along the x-axis (rightwards for positive, leftwards for negative), then 'y' units vertically along the y-axis (upwards for positive, downwards for negative). For instance, point (3,2) is reached by moving 3 units to the right and 2 units up from the origin.

Step 1: Understanding Cartesian Coordinates

On a rectangular coordinate system, a point is defined by an ordered pair of numbers called coordinates. The ordered pair of coordinates is written as \((x, y)\), where 'x' refers to the horizontal distance from the origin (measured along the x-axis), and 'y' refers to the vertical distance from the origin (measured along the y-axis). The origin is the point where the x and y axes intersect, and is defined as \((0,0)\). The x coordinate can be negative (to the left of the origin) or positive (to the right of the origin). Similarly, the y coordinate can be negative (below the origin) or positive (above the origin).

Step 2: Locating and Plotting the Point

To plot a point on a rectangular coordinate system, first look at the x coordinate and start at the origin. If the x-coordinate is positive, move that number of units to the right; if it’s negative, move to the left. Then look at the y coordinate. If the y-coordinate is positive, move up from the x position; if it’s negative, move down. The intersection of the x and y paths is the position of the point. For example, to plot the point (3,2), you would start at the origin, move three units to the right (as 3 is positive), and then move up two units (as 2 is positive). Your point (3,2) would be located there.

Step 3: Visualization of the Point (3,2)

After moving 3 units right along the x-axis and 2 units up along the y-axis, you can plot a mark. Label this mark with the coordinates \((3, 2)\). Therefore, if a graph is present, the dot labeled (3,2) would be the visual representation of this point in the Cartesian coordinate system.