Question

Sketch the following sets of points in the \(x-y\) plane. $$ \\{(x, y): x=2, y \in[0,1]\\} $$

Short answer

The points in the set \(\{(x, y): x=2, y \in [0,1]\}\) form a vertical line segment on the \(x-y\) plane, starting from the point \((2,0)\) and ending at the point \((2,1)\).

Step-by-step solution

Understanding the coordinates

The given set includes all points where \(x = 2\) and \(y\) is in the range between 0 and 1, inclusive. This means that for any point in the given set, the x-coordinate is always 2, while the y-coordinate can be any value from 0 to 1. This will form a vertical line segment on the plane.

Setting up the graph

Draw a coordinate plane labelled with x and y axes. On the x-axis, label a point as 2. This represent the x-coordinate of the points in the set.

Plotting the points

From the point labelled 2 on the x-axis, draw a vertical line upwards until you reach 1 on the y-axis (inclusive). Then, stop and go downwards until you reach 0 on the y-axis. This vertical line represents all points in the set where the x-coordinate is 2 and the y-coordinate varies from 0 to 1 inclusive.

Key concepts

Plotting Points on the Coordinate Plane

Understanding how to plot points is fundamental in creating a visual representation of mathematical concepts on the coordinate plane. The coordinate plane consists of two perpendicular lines, traditionally called axes, which are used to define the location of points in a two-dimensional space. The horizontal line is known as the x-axis, while the vertical line is called the y-axis.

To plot a point, you need two coordinates: an x-coordinate which tells you how far along the point is horizontally, and a y-coordinate which tells you how far up or down the point is vertically. These coordinates are often written in the form \(x, y\), and they define the exact location of the point on the graph. For example, the coordinates \(2, 0\) mean the point is located 2 units to the right of the origin along the x-axis, and on the y-axis it is at the origin.

Practice With Coordinates

To get comfortable with plotting points, start with locating each integer on both the x-axis and y-axis. Then, practice by plotting points provided in exercises, like \(2, 1\) or \(2, 0\). Always begin by moving along the x-axis, then move up or down to reach the final y-value.

Understanding Vertical Line Segments

A vertical line segment on the coordinate plane is one that runs up and down, parallel to the y-axis, and it has a constant x-value. In the context of our exercise, the set \(\{(x, y): x=2, y \in[0,1]\}\) describes a vertical line segment because for all points in this set, the x-coordinate is fixed at 2 while the y-coordinate ranges from 0 to 1.

Visually, this would appear as a line starting at \(2, 0\) and ending at \(2, 1\) because these coordinates mark the endpoints of the segment within that range. This segment shows the relation where all points share a common x-value. It is useful in depicting rates of change in one dimension (in this case, the y dimension), boundaries, and constraints in equations or inequalities.

Drawing Vertical Lines

When you're asked to draw such a line, first plot the two endpoints at \(2, 0\) and \(2, 1\). Then, use a ruler to connect these points with a straight line. The line should intersect the x-axis exactly at x=2 to ensure it's vertical. Remember, in a vertical line, the y-values change but the x-value remains constant.

The X-Y Axis

The x-y axis is arguably the most important feature in a coordinate plane. Together, they divide the plane into four quadrants and allow us to pinpoint the location of points in two dimensions. The point where they intersect is referred to as the origin, which has coordinates \(0, 0\).

The x-axis is horizontal with positive values to the right and negative values to the left of the origin. The y-axis is vertical with positive values above the origin and negative values below it. Each axis is a straight line extending infinitely in both directions, and they are mutually perpendicular to each other.

Significance in Graphing

The axes serve as reference lines from which measurements are made and thus, they enable us to graph equations and inequalities. For instance, the equation \(x=2\) is represented graphically by a vertical line intersecting the x-axis at 2, which is what our given exercise requires us to draw. Understanding the structure and layout of the x-y axis is key to mastering more complex graphing challenges.