Problem 17 - 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. \( x = -4 \) and \( y > 0 \)

Step-by-Step Solution

Answer

The point (x, y) is located in Quadrant II.

Step 1: Identify the Values of x and y

First, let's look at the conditions given. The exercise specifies that x = -4 and y > 0. This means that the value of x is negative and the value of y is positive.

Step 2: Check the Quadrant Conditions

Since x is negative and y is positive, this fits the conditions for Quadrant II, where x < 0 and y > 0.

Step 3: Conclusion

Therefore, given the conditions x = -4 and y > 0, the point (x, y) must be located in Quadrant II.

Key Concepts

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which points are plotted. It consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants, each having different sign characteristics for coordinates. The point where the axes intersect is known as the origin, labeled (0,0).

The coordinate plane makes it easier to visualize and understand mathematical relationships and geometric properties by representing points as coordinate pairs. Some key aspects to remember:

• Each point on the plane is expressed as ewline (x, y), where x is the horizontal distance from the origin, and y is the vertical distance.
• The easiest way to identify a position on the coordinate plane is by using the (x, y) format, where 'x' denotes the value on the x-axis and 'y' denotes the value on the y-axis.
• Quadrants are labeled I, II, III, and IV starting from the top right and moving counter-clockwise.

Understanding the coordinate plane is fundamental in both algebra and geometry, helping in the visualization and solving of equations.

Negative x-values

Negative x-values are simply the values of x on the left side of the y-axis in the coordinate plane. These values are crucial for determining which quadrant a point falls into, given its y-value. Here's what you need to remember about negative x-values:

• When x is negative, it indicates a position left of the origin on the x-axis.
• Points with negative x-values can be found in Quadrants II and III, depending on the sign of their y-value.
• The transition from positive to negative x-values happens at the origin (0 on the x-axis).

For a negative x-coordinate, you would typically subtract from the right side towards the left side of the plane. In exercises like the given example, understanding that x = -4 places the point towards the left half of the plane helps in determining the point's quadrant location.

Positive y-values

Positive y-values indicate the vertical position of points above the x-axis in the coordinate plane. Understanding where these values can be seen is integral to identifying points on the plane:

• When y is positive, it means the point lies above the x-axis.
• Points with positive y-values exist in Quadrants I and II.
• The transition from negative to positive y-values happens at the origin (0 on the y-axis).

In our exercise, knowing that y > 0 confirms that any point with this property will be located above the x-axis. Combined with an x-value that is negative, such as x = -4, positive y-values confirm the point's placement in Quadrant II. This understanding is critical when positioning any coordinate in the context of a coordinate plane.