Question

Plot each point in the xy-plane. State which quadrant or on what coordinate axis each point lies. Plot the points \((0,3),(1,3),(-2,3),(5,3),\) and \((-4,3) .\) Describe the set of all points of the form \((x, 3),\) where \(x\) is a real number.

Short answer

(0,3) on y-axis; (1,3) and (5,3) in Quadrant I; (-2,3) and (-4,3) in Quadrant II. Set of points \((x, 3)\) lies on line \y = 3\.

Step-by-step solution

- Understand the Points

Each point is given in the form \(x, y\). Here, all points share the same y-coordinate, 3.

- Plot the Points

Place each point \((0,3), (1,3), (-2,3), (5,3), (-4,3)\) on the xy-plane.

- Identify Quadrants and Axes

* (0,3) lies on the y-axis. * (1,3) lies in Quadrant I. * (-2,3) lies in Quadrant II. * (5,3) lies in Quadrant I. * (-4,3) lies in Quadrant II.

- Describe the Set of All Points

Points of the form \((x, 3)\) where \x\ is a real number lie on the horizontal line \y = 3\. These points form a horizontal line that extends infinitely in both directions along the x-axis at the level where \y = 3\.

Key concepts

xy-plane

When we talk about the xy-plane, we are referring to a two-dimensional surface defined by two perpendicular lines, known as the x-axis and y-axis. The xy-plane allows us to locate points using a pair of numerical coordinates.
Each point on this plane is identified by an ordered pair \((x, y)\). The x-coordinate tells us how far to move left or right, while the y-coordinate tells us how far up or down to go.

To visualize this, imagine graph paper. The horizontal line is the x-axis and the vertical line is the y-axis. The point where they intersect is called the origin, marked as \(0,0\).

coordinate system

The coordinate system is a method used to determine the position of points. In the xy-plane, it uses two numbers, the x-coordinate and the y-coordinate, to describe a point's location.

Think of the coordinate system like the map of a city. The x-axis can be the main street running horizontally, and the y-axis can be the major avenue running vertically. Every intersection in the city could be described using a pair of numbers, such as Street 2, Avenue 3 \((2, 3)\).

The coordinate system makes it easier to plot points, understand their relationships, and navigate through the xy-plane efficiently.

quadrants

The xy-plane is divided into four regions called quadrants. Each quadrant is determined by the signs of x and y coordinates.

- **Quadrant I**: Both x and y coordinates are positive (e.g., \(1, 3\)).
- **Quadrant II**: The x-coordinate is negative, and the y-coordinate is positive (e.g., \(-2, 3\)).
- **Quadrant III**: Both x and y coordinates are negative.
- **Quadrant IV**: The x-coordinate is positive, and the y-coordinate is negative.

The quadrants help quickly identify the general area of a point within the xy-plane, making it easier to comprehend and plot.

axes

The axes consist of the x-axis and the y-axis which intersect at the origin \(0, 0\). The x-axis runs horizontally through zero, to the right is positive and to the left is negative.

The y-axis runs vertically through zero, upwards is positive and downwards is negative. These two axes create the framework for the coordinate system and allow for precise plotting of points.

If a point lies on either axis, it signifies that one of its coordinates is zero. For instance, the point \((0, 3)\) lies on the y-axis.

horizontal line

A horizontal line runs straight left to right, parallel to the x-axis. Points forming a horizontal line share the same y-coordinate.

For example, in the given problem, all points like \((x, 3)\) lie on the horizontal line where \(y = 3\). This is because no matter the value of x, the y-coordinate remains consistent.

This forms a straight, infinite line across the xy-plane, indicating consistent height but varying horizontal position.