Problem 36 - A Graphical Approach to College Algebra

Question

Name the possible quadrants in which the point ( \(x, y\) ) can lie if the given condition is true. $$x y<0$$

Step-by-Step Solution

Answer

The point can lie in Quadrant II or Quadrant IV.

Step 1: Understanding the Product Condition

The condition given is \(xy < 0\). This means that the product of \(x\) and \(y\) is less than zero. In other words, \(x\) and \(y\) must have opposite signs.

Step 2: Identifying Quadrants with Opposite Signs

In the coordinate plane, quadrants are defined as follows: - Quadrant I: \((+x, +y)\), both positive - Quadrant II: \((-x, +y)\), \(x\) negative and \(y\) positive - Quadrant III: \((-x, -y)\), both negative - Quadrant IV: \((+x, -y)\), \(x\) positive and \(y\) negative For \(xy < 0\), we need these points to have opposite signs:

Step 3: Identifying Quadrant II as a Possible Location

In Quadrant II, the coordinates \((-x, +y)\) mean \(x\) is negative and \(y\) is positive. The product \(xy\) will be negative because a negative times a positive is negative.

Step 4: Identifying Quadrant IV as a Possible Location

In Quadrant IV, the coordinates \((+x, -y)\) mean \(x\) is positive and \(y\) is negative. Again, the product \(xy\) will be negative because a positive times a negative is negative.

Key Concepts

Quadrants

In the coordinate plane, there are four distinct regions known as quadrants. These quadrants are defined by the positive and negative axes of the x-axis and y-axis. Understanding the properties of these quadrants is fundamental in solving many mathematical problems.

• Quadrant I: Located in the top right position, where both the x-coordinate and y-coordinate are positive \((+x, +y)\).


• Quadrant II: Found in the top left, where the x-coordinate is negative and the y-coordinate is positive \((-x, +y)\).


• Quadrant III: Located in the bottom left, where both the x-coordinate and y-coordinate are negative \((-x, -y)\).


• Quadrant IV: Situated in the bottom right, where the x-coordinate is positive and the y-coordinate is negative \((+x, -y)\).

As you can see, the sign of the coordinates defines which quadrant a point belongs to. Recognizing how these quadrants are structured helps us determine where points with specific properties, such as negative or positive products, will be located.

Signs of Coordinates

The sign of a coordinate in the Cartesian plane tells us much about its position. This is incredibly useful in predicting the behavior of points and solving problems related to coordinates. The sign is determined as follows:

• The x-coordinate is positive if it lies to the right of the y-axis and negative if it lies to the left.


• The y-coordinate is positive if it is above the x-axis and negative if it is below.

Understanding the signs of the coordinates allows us to quickly hypothesize which quadrants a coordinate may lie in. For instance, knowing that one of the coordinates is positive and the other is negative, it is clear that such a point must lie either in Quadrant II or Quadrant IV. The concept of coordinate signs is a critical part of analyzing their interactions, especially when considering the product of these coordinates.

Product of Coordinates

The product of coordinates refers to multiplying the x-coordinate by the y-coordinate of a point in the Cartesian plane. This product is an important concept when analyzing the relationships between coordinate points.

When the product of the coordinates \((x \, y)\) of a point is less than zero \((xy < 0)\), it indicates that the coordinates have opposite signs. This is because:

• A positive number multiplied by a negative number will always yield a negative product.


• Similarly, a negative number multiplied by a positive one results in a negative product.

Thus, if the condition \(xy < 0\) holds, the point must be located in either Quadrant II, where \((-x, +y)\) occur or in Quadrant IV, where \((+x, -y)\) are evident. This understanding helps in pinpointing the location of points based on product conditions, making it a powerful tool in coordinate geometry.