Problem 9 - Contemporary Precalculus

Question

(a) If the first coordinate of a point is greater than 3 and its second coordinate is negative, in what quadrant does it lie? (b) What is the answer in part (a) if the first coordinate is less than \(3 ?\)

Step-by-Step Solution

Answer

And in which quadrant would it lie if the x-value is less than 3, while still having a negative y-value? Answer: If the x-value is greater than 3 and the y-value is negative, the point lies in the fourth quadrant. If the x-value is less than 3 and the y-value is negative, the point lies in the third quadrant.

Step 1: Solution for scenario 1

Since the x-value is greater than 3 and the y-value is negative, the point lies in the fourth quadrant, where x > 0 and y < 0.

Step 2: Solution for scenario 2

In this case, the x-value is less than 3 (x < 3) and the y-value is still negative. So, the point lies in the third quadrant, where x < 0 and y < 0.

Key Concepts

Quadrants

When we talk about the Cartesian Coordinate System, we are referring to a two-dimensional plane where points are located using pairs of numbers known as coordinates. This plane is divided into four sections known as quadrants.
The quadrants are created by the intersection of two axes:

• The horizontal axis, known as the x-axis.
• The vertical axis, known as the y-axis.

These axes divide the plane into four quadrants.
Each quadrant is a square section identified using roman numerals:

• Quadrant I: Located in the top-right where both x and y coordinates are positive ( quad x > 0, y > 0 ).
• Quadrant II: Found in the top-left where x is negative and y is positive ( quad x < 0, y > 0 ).
• Quadrant III: In the bottom-left where both x and y coordinates are negative ( quad x < 0, y < 0 ).
• Quadrant IV: Situated in the bottom-right where x is positive and y is negative ( quad x > 0, y < 0 ).

Understanding how these quadrants work is key to correctly placing any point on the coordinate plane.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a grid used to visually place and locate points. It's made up of two intersecting number lines:

• The x-axis, which runs horizontally.
• The y-axis, which runs vertically.

At the intersection of these two axes is the origin, marked as (0, 0).
The plane extends infinitely in all directions from the origin, though it is commonly shown in a set window around the center.
This plane is a fundamental concept because it gives us a method to translate algebraic expressions into visual images, and to see the relationship between numbers. By using the coordinate plane, mathematical problems can be made more intuitive through visualization.

Coordinates

Coordinates are a pair of numbers that determine the precise location of a point on the coordinate plane. Each coordinate is expressed as ( x, y ), combining:

• x -value: Represents the horizontal location. Positive during right straightforward shoe on the x-axis, negative shifts to the left.
• y -value: Represents the vertical position. Positive when moving upward, negative when the point goes downward.

Together, these numbers specify a unique point on the plane. These points could be anywhere in one of the four quadrants, on the axes, or at the origin. Getting comfortable with reading and plotting these pairs leads to a deeper mathematical understanding and capability to solve related problems.
Being familiar with how combinations of positive and negative numbers affect direction on the plane is a basic skill in coordinate geometry.