Problem 71 - Algebra A Combined Function

Question

For Exercises 71 through 75, fill in each blank with "0," "positive," or "negative." For Exercises 76 and 77, fill in each blank with "x"or "y." Point \(\quad\) (_____,_____) Location \(\quad\) quadrant III

Step-by-Step Solution

Answer

(negative, negative)

Step 1: Understand Quadrant III

In the coordinate plane, Quadrant III is where both x and y coordinates are negative. This means any point in Quadrant III will have a form (-x, -y) where x and y are both positive values, but the point itself is defined with negative values.

Step 2: Fill in the Blank for the X-coordinate

Since Quadrant III necessitates a negative x-coordinate, write 'negative' for the x-coordinate location in the blank.

Step 3: Fill in the Blank for the Y-coordinate

Similarly, in Quadrant III, the y-coordinate is also negative. Consequently, fill in 'negative' for the y-coordinate location in the blank.

Key Concepts

Understanding Quadrants

In coordinate geometry, a quadrant refers to one of the four sections of the coordinate plane.
These sections are created by the intersection of the x-axis and y-axis, which divide the plane into four equal parts.
This division helps in easily identifying the location of points on the plane. Let's break down the quadrants:

• Quadrant I: Both x and y coordinates are positive. Any point here has the form (positive, positive).
• Quadrant II: Here, the x coordinate is negative while the y coordinate is positive, forming points like (negative, positive).
• Quadrant III: Both x and y coordinates are negative. Therefore, any point here appears as (negative, negative).
• Quadrant IV: The x coordinate is positive, and the y coordinate is negative, so points look like (positive, negative).

Knowing these characteristics helps us to quickly determine and identify the nature of points based on their position on the plane.

The Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface formed by two perpendicular lines called the x-axis and the y-axis.
The point of intersection of these axes is the origin, marked as (0,0), which is the starting point for all measurements in this system. Let's explore the components of the coordinate plane:

• x-axis: This horizontal line runs left to right. Positive values are to the right of the origin, while negative values are to the left.
• y-axis: This vertical line runs up and down. Positive values are above the origin, and negative values are below it.
• Origin: The intersection point (0,0) where the coordinate axes meet.
• Coordinates: Each point on the plane is described by an ordered pair (x, y), indicating its position relative to the origin.

Using the coordinate plane helps us in graphically representing equations and analyzing relationships between variables.

Negative Coordinates

Negative coordinates indicate positions that are either to the left of the y-axis or below the x-axis, or both, depending on whether the x or y coordinate is negative.
Understanding negative coordinates is important as it expands our ability to navigate and utilize all parts of the coordinate plane efficiently. Consider these situations:

• Negative x-coordinate: Points are positioned to the left of the origin. For example, (-2, 3) is in the second quadrant because the x-value is negative.
• Negative y-coordinate: Points are positioned below the origin. For instance, (4, -3) is in the fourth quadrant where the y-value is negative.
• Both coordinates negative: This places the point in the third quadrant where both x and y values are negative, such as (-5, -7).

Having a clear understanding of negative coordinates helps in effectively solving problems related to locating and understanding points on the coordinate plane.