Problem 1 - Algebra and Trigonometry

Question

Plot the point whose rectangular coordinates are (3,-1) . What quadrant does the point lie in?

Step-by-Step Solution

Answer

Quadrant IV

Step 1: Identify the coordinates

Locate the given rectangular coordinates, which are (3, -1). This means the x-coordinate is 3 and the y-coordinate is -1.

Step 2: Plot the point on the coordinate plane

Start at the origin (0,0). Move 3 units to the right along the x-axis to account for the x-coordinate. Then, move 1 unit down along the y-axis because the y-coordinate is -1. Mark this point on the graph.

Step 3: Determine the quadrant

Recall the quadrants on the coordinate plane: Quadrant I (+,+), Quadrant II (-,+), Quadrant III (-,-), Quadrant IV (+,-). Since the x-coordinate is positive and the y-coordinate is negative, the point (3, -1) lies in Quadrant IV.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are ordered pairs of numbers that define the position of a point on a coordinate plane.
The two numbers in the pair are called the x-coordinate and the y-coordinate.
The x-coordinate tells us how far to move horizontally from the origin (0,0), while the y-coordinate tells us how far to move vertically.
For example, in the point (3, -1), 3 is the x-coordinate and -1 is the y-coordinate.
This means you move 3 units to the right and 1 unit down from the origin to plot the point.
Understanding rectangular coordinates is essential for graphing and analyzing points on a coordinate plane.

Coordinate Plane Quadrants

The coordinate plane is divided into four sections called quadrants.
These quadrants help us understand the position of points based on the signs of their coordinates.
Here is a quick breakdown of each quadrant:

• Quadrant I: Both x and y are positive (+,+).
• Quadrant II: x is negative, and y is positive (-,+).
• Quadrant III: Both x and y are negative (-,-).
• Quadrant IV: x is positive, and y is negative (+,-).

When you have a point like (3, -1), you look at the signs: x is positive and y is negative.
This means the point is in Quadrant IV.
Remembering these quadrants helps in quickly determining the location of any point on the coordinate plane.

Graphing Points

Graphing points on a coordinate plane involves a few simple steps.
First, you start at the origin (0,0).
Then, you move horizontally along the x-axis by the amount given in the x-coordinate.
After that, you move vertically along the y-axis by the amount given in the y-coordinate.
Here's an example:

• For the point (3, -1), begin at (0,0).
• Move 3 units to the right because the x-coordinate is 3.
• Then move 1 unit down because the y-coordinate is -1.

Mark the point where you land.
That’s how easy it is to graph points!
Practicing this will make plotting points second nature.