Problem 16 - Precalculus with Limits

Question

In Exercise 15-24, determine the quadrant(s) in which \( (x, y) \) is located so that the condition(s) is (are) satisfied. \( x < 0 \) and \( y < 0 \)

Step-by-Step Solution

Answer

The point (x, y) is in Quadrant III.

Step 1: Understanding the Coordinate Plane

The coordinate plane is divided into four quadrants by the x-axis and the y-axis. Quadrant I is where both x and y are positive. Quadrant II is where x is negative and y is positive. Quadrant III is where both x and y are negative. Quadrant IV is where x is positive and y is negative.

Step 2: Applying the given conditions

The conditions given are \(x < 0\) and \(y < 0\). This means that both x and y must be less than zero. Bearing in mind the information from Step 1, one can conclude that the point (x, y) is in the quadrant where both x and y are negative.

Step 3: Determining the Quadrant

Based on Step 2, since both x and y are negative, our point (x, y) must be in Quadrant III.

Key Concepts

Quadrants

A coordinate plane is divided into four sections called quadrants. Each quadrant helps us identify where a point lies based on its coordinates. The quadrants are numbered using Roman numerals I, II, III, and IV, and they follow a counter-clockwise direction starting from the upper right.

Here is a breakdown of each quadrant:

• Quadrant I: Both coordinates are positive \(x > 0\) and \(y > 0\).
• Quadrant II: The x-coordinate is negative while the y-coordinate is positive \(x < 0\) and \(y > 0\).
• Quadrant III: Both coordinates are negative \(x < 0\) and \(y < 0\).
• Quadrant IV: The x-coordinate is positive while the y-coordinate is negative \(x > 0\) and \(y < 0\).

When you need to place a point on this plane, understand which quadrant it belongs to based on the signs of x and y. This is vital in both visualizing and solving problems in coordinate geometry.

Negative Coordinates

Negative coordinates occur when either of the x or y values of a point is less than zero. In real-life scenarios, negative coordinates might represent specific conditions, like spending more money than you have or traveling in the opposite direction along a route.

In the context of our coordinate plane:

• A negative x-coordinate means the point is to the left of the y-axis.
• A negative y-coordinate indicates the point is below the x-axis.

For the given conditions where both x and y are negative, this aligns perfectly with Quadrant III. When dealing with problems that feature negative values, always visualize the placement of these points, imagining how far left and down they might sit from the origin (0,0). This helps in confirming their quadrant placement.

Coordinate Plane

The coordinate plane acts as a grid for graphing and solving equations, consisting of two perpendicular lines that intersect to form right angles. These lines are known as the x-axis (horizontal) and y-axis (vertical), and their intersection point is called the origin, marked as \(0,0\). Each point in this plane is identified using a pair \((x, y)\) which represents its position relative to these axes.

Understanding how points are mapped onto the coordinate plane allows us to apply the rules of quadrants and sign determination.

• If both x and y are positive, head to Quadrant I.
• For a negative x and positive y, look to Quadrant II.
• Negative values for both x and y send the point to Quadrant III.
• A positive x with a negative y finds a home in Quadrant IV.

The setup of the coordinate plane makes it an essential tool for plotting points, solving equations, and visually representing relationships between variables. Practice navigating this plane to boost your proficiency in coordinate geometry.