Problem 78 - Algebra A Combined Function

Question

Give an example of an ordered pair whose location is in (or on) a. quadrant I b. quadrant II c. quadrant III d. quadrant IV e. \(x\) -axis f. \(y\) -axis

Step-by-Step Solution

Answer

Quadrant I: (2, 3); Quadrant II: (-4, 5); Quadrant III: (-3, -1); Quadrant IV: (6, -2); x-axis: (5, 0); y-axis: (0, 7).

Step 1: Identify Quadrant I

A point in Quadrant I has both positive x and y coordinates. For example, the ordered pair \((2, 3)\) lies in Quadrant I.

Step 2: Identify Quadrant II

A point in Quadrant II has a negative x coordinate and a positive y coordinate. For example, the ordered pair \((-4, 5)\) is located in Quadrant II.

Step 3: Identify Quadrant III

A point in Quadrant III has both negative x and y coordinates. For example, the ordered pair \((-3, -1)\) lies in Quadrant III.

Step 4: Identify Quadrant IV

A point in Quadrant IV has a positive x coordinate and a negative y coordinate. An example of such a pair is \((6, -2)\).

Step 5: Locate on x-axis

A point on the x-axis has a zero y coordinate. For instance, \((5, 0)\) is located on the x-axis.

Step 6: Locate on y-axis

A point on the y-axis has a zero x coordinate. An example is \((0, 7)\), which lies on the y-axis.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can locate points using ordered pairs. These pairs consist of two numbers written in the form \(x, y\). This grid-like system helps us to easily identify the exact position of any point in space by specifying its horizontal and vertical distance from the origin.

• The horizontal axis is called the x-axis.
• The vertical axis is known as the y-axis.

The point where these axes intersect is called the origin, represented by the ordered pair \(0, 0\). This setup allows us to navigate and plot points with precision, making the coordinate plane a fundamental tool in mathematics.

Quadrants

The coordinate plane is divided into four sections, known as quadrants. Each quadrant is labeled with Roman numerals and has distinct characteristics based on the signage of the x and y values. Here's a breakdown:

• Quadrant I: Both x and y coordinates are positive.
• Quadrant II: The x coordinate is negative, while the y coordinate is positive.
• Quadrant III: Both x and y coordinates are negative.
• Quadrant IV: The x coordinate is positive, and the y coordinate is negative.

This division into quadrants helps in systematically analyzing and categorizing points on the coordinate plane. For example, the point \(2, 3\) lies in Quadrant I, indicating its position falls where both coordinates are positive.

X-Axis and Y-Axis

In the coordinate system, the x-axis and y-axis are the two main reference lines used to plot points. The x-axis runs horizontally, while the y-axis runs vertically, intersecting at the origin \(0, 0\). Together, they form the framework of the coordinate plane.

• The x-axis is the boundary between Quadrants I & IV on the top and II & III below.
• The y-axis is the boundary between Quadrants I & II on the right and III & IV on the left.

Points that lie directly on the x-axis will have a y-coordinate of zero, such as \(5, 0\). Conversely, points on the y-axis have an x-coordinate of zero, like \(0, 7\). This makes it easy to identify which axis a point lies on simply by looking at these zeroes.

Positive and Negative Coordinates

Knowing whether a coordinate is positive or negative helps in determining its location on the coordinate plane. Coordinates are labeled as positive or negative based on their relationship to the origin.

• Positive x-coordinates are to the right of the y-axis.
• Negative x-coordinates are to the left of the y-axis.
• Positive y-coordinates are above the x-axis.
• Negative y-coordinates are below the x-axis.

Understanding this concept of positive and negative coordinates allows us to predict where a point will lie on the plane. For instance, the ordered pair \((-4, 5)\) resides in Quadrant II due to the negative x and positive y values.