Question

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

Short answer

The points lie on the line x = 2. Point (2,0) is on the x-axis, (2,-3) and (2,-1) are in Quadrant IV, (2,1) and (2,4) are in Quadrant I.

Step-by-step solution

- Understand the points

The given points are \(2,0\), \(2,-3\), \(2,4\), \(2,1\), and \(2,-1\). In each of these points, the x-coordinate is 2, which implies all points are aligned vertically on the x = 2 line.

- Plot each point

On a Cartesian plane, plot each of the given points:\begin{itemize}\item \(2,0\) is on the x-axis.\item \(2,-3\) is 3 units down from the x-axis.\item \(2,4\) is 4 units up from the x-axis.\item \(2,1\) is 1 unit up from the x-axis.\item \(2,-1\) is 1 unit down from the x-axis.\end{itemize}\These points will all lie on a vertical line at x = 2.

- Identify quadrants or axes

Determine the position of each point:\begin{itemize}\item \(2,0\) lies on the x-axis.\item \(2,-3\) and \(2,-1\) lie in Quadrant IV.\item \(2,1\) and \(2,4\) lie in Quadrant I.\end{itemize}

- Describe the set of all points

The set of all points of the form \(2, y\) where \(y\) is a real number is a vertical line passing through (2, y) on the Cartesian plane. This line has all the points with an x-coordinate of 2.

Key concepts

Plotting Coordinates

Plotting coordinates on the Cartesian plane is a fundamental skill in math. Each point on the plane is defined by a pair of numbers, known as coordinates, written as \(x, y\). The first number \(x\) represents the horizontal position, and the second number \(y\) represents the vertical position. To plot a point like \(2, -3\), find the value on the x-axis, which is 2. Then, move vertically to -3 on the y-axis. Mark the point where these two meet. Repeat for each coordinate to see the points on the plane.

Quadrants

The Cartesian plane is divided into four sections, or quadrants, by the x-axis and y-axis. Each quadrant is identified by Roman numerals:

• **Quadrant I (First Quadrant)**: Both x and y are positive.
• **Quadrant II (Second Quadrant)**: x is negative and y is positive.
• **Quadrant III (Third Quadrant)**: Both x and y are negative.
• **Quadrant IV (Fourth Quadrant)**: x is positive and y is negative.

Points on the axes do not belong to any quadrant but to the axis they lie on. For example, \(2, 0\) lies on the x-axis. Points like \(2, 1\) and \(2, 4\) lie in Quadrant I as their y-values are positive. Points like \(2, -3\) and \(2, -1\) are in Quadrant IV because their y-values are negative.

Vertical Lines on Cartesian Plane

A vertical line on the Cartesian plane means that all its points have the same x-coordinate. For example, all points of the form \(2, y\) lie on a vertical line where the x-coordinate is fixed at 2. This line can extend infinitely both up and down. The y-coordinate can be any real number, which means the points can be anywhere vertically along this line. To visualize, if you plot \(2, -3\), \(2, 0\), \(2, 1\), \(2, 4\), and more, you'll see they all line up vertically. This concept is crucial because understanding vertical and horizontal lines helps analyze and interpret different types of graphs.