Problem 73 - 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 IV

Step-by-Step Solution

Answer

(positive, negative)

Step 1: Understanding Quadrant IV

In the Cartesian coordinate system, Quadrant IV is the bottom-right section. Here, the x-values are positive and the y-values are negative because it lies to the right of the y-axis and below the x-axis.

Step 2: Identifying Coordinates in Quadrant IV

Since Quadrant IV has positive x-values and negative y-values, when a point is located in this quadrant, its x-coordinate is 'positive' and its y-coordinate is 'negative'.

Step 3: Filling in the Blanks

The exercise asks for the nature of x and y values for a point in Quadrant IV. Based on the understanding of the coordinates in Quadrant IV: - The x-coordinate will be 'positive' because it is to the right of the y-axis. - The y-coordinate will be 'negative' because it is below the x-axis.

Key Concepts

Quadrant IV

Quadrant IV is one of the four sections in the Cartesian coordinate system. Imagine a flat surface divided into four regions using two intersecting lines: the vertical y-axis and the horizontal x-axis. These axes create four quadrants which are traditionally numbered counter-clockwise, starting from the top right.

Quadrant IV is situated at the bottom right of the coordinate plane. This means it's to the right of the y-axis and below the x-axis. Because of this positioning, certain characteristics define it:

• The x-coordinates in this quadrant are positive since all points lie to the right of the y-axis.
• The y-coordinates are negative because the quadrant is below the x-axis.

Understanding the layout of Quadrant IV is vital for graphing points accurately and interpreting which values x and y should have in that space.

x-coordinate

In the Cartesian coordinate system, an x-coordinate is one of the two values used to indicate the position of a point. It's generally represented on the horizontal axis. The x-coordinate tells us how far left or right a point is relative to the origin, which is located at (0,0).

When identifying the x-coordinate of a point in Quadrant IV:

• It will always be positive. This is because locations in this quadrant are positioned to the right of the vertical y-axis.
• Being positive implies that the value is greater than zero, showing a movement to the right on the x-axis from the center point or origin.

Understanding x-coordinates helps in plotting points correctly along the horizontal direction of a graph, which is crucial when determining the exact position of any point on the Cartesian plane.

y-coordinate

The y-coordinate in a Cartesian coordinate system represents the vertical position of a point and is expressed along the y-axis. It specifies how far above or below a point is from the origin, where both axes intersect at (0,0).

For points situated in Quadrant IV:

• The y-coordinate is always negative. This is due to the region sitting below the horizontal x-axis.
• Negative y-values mean that the point droops below the origin, indicating a downward placement from the horizontal center of the graph.

By understanding the role of y-coordinates, one can accurately visualize and locate points in any quadrant, including Quadrant IV, making it easier to work with graphing and coordinate geometry.