Problem 8 - Algebra and Trigonometry

Question

In Exercises 1–12, plot the given point in a rectangular coordinate system. $$ (3,-2) $$

Step-by-Step Solution

Answer

The point (3,-2) is located 3 units to the right of the origin and 2 units down from there.

Step 1: Identify the x and y coordinates

The point given is (3,-2). The first value is the horizontal (x) coordinate, and the second value is the vertical (y) coordinate. So, the x-coordinate is 3, and the y-coordinate is -2.

Step 2: Plot the point on the graph

Starting at the origin, move 3 units to the right for x = 3. Then go down 2 units for y = -2. This is the position of the point (3,-2) in the rectangular coordinate system. Draw a dot here to represent the point (3, -2).

Key Concepts

Rectangular Coordinate System

Understanding the rectangular coordinate system is fundamental for graphing points. It is a two-dimensional plane consisting of a horizontal axis, known as the x-axis, and a vertical axis, called the y-axis. These axes intersect at a point called the origin, which has coordinates \(0,0\). The entire system is based on a grid, and each point on this grid is defined by a pair of numbers. These numbers represent the distances from the origin along the x and y axes, respectively.

The system is also divided into four quadrants by the axes: the first quadrant (I) where both x and y coordinates are positive, the second quadrant (II) with a negative x and a positive y, the third quadrant (III) with both x and y being negative, and the fourth quadrant (IV) with a positive x and a negative y. This arrangement allows you to locate every possible point based on its positive or negative values on the axes.

X and Y Coordinates

In a rectangular coordinate system, every point is defined by an ordered pair of numbers, commonly referred to as coordinates. The first number of the pair is the x-coordinate, and it indicates the distance the point is from the origin along the x-axis, moving right for positive values and left for negative values. The second number is the y-coordinate, which shows the distance from the origin along the y-axis, moving up for positive values and down for negative values.

Understanding Coordinates:

• The x-coordinate is always listed first in the ordered pair.
• The y-coordinate is always listed second in the ordered pair.
• Positive x-coordinates extend to the right of the origin, while negative x-coordinates extend to the left.
• Positive y-coordinates extend upwards from the origin, while negative y-coordinates extend downwards.

Using coordinates, we can precisely pinpoint the location of a point on the plane, ensuring accuracy in both mathematics and real-world mapping.

Graphing on a Cartesian Plane

Graphing on a Cartesian plane involves placing points on the grid according to their x and y coordinates. This process begins at the origin of the axes. Here's a simple way to graph the point \(3,-2\):

Step-by-Step Graphing:

Firstly, start at the origin where \(x = 0\) and \(y = 0\). Move horizontally to the right if the x-coordinate is positive, or to the left if it is negative. For \(3,-2\), you'd move 3 units to the right because the x-coordinate is 3. Next, move vertically up if the y-coordinate is positive, or down if it's negative. In this case, you'd move 2 units down since the y-coordinate is -2. Place a dot or other marker to indicate the point on the grid.

Remember, with practice, graphing becomes intuitive. You'll be able to visualize a point's location with ease once you understand the relationship between the coordinates and their positions on the Cartesian plane.