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Magazine Chapter 2: Problem 21
\(19-32\) Sketch the region given by the set. $$ \\{(x, y) | x=3\\} $$
Short Answer
Expert verified
Sketch a vertical line at \(x=3\) extending across the full range of y-values.
Step by step solution
01
Understand the Equation
The equation given is \(x = 3\). This tells us that for every point in the set, the x-coordinate is fixed at 3. The y-coordinate can be any real number.
02
Identify the Type of Line
Since \(x = 3\) is an equation of the form \(x = c\), where \(c\) is a constant, this represents a vertical line in the coordinate plane.
03
Plot the Line on Coordinate Plane
To sketch the region, plot the vertical line that passes through all points where the x-coordinate is 3. This line will be parallel to the y-axis.
04
Cover all Possible y-values
Draw the vertical line extending in both directions, up towards positive infinity and down towards negative infinity on the y-axis. This shows that the y-coordinate can take any value.
05
Label the Line
Ensure the line is labeled as \(x = 3\) on the graph to indicate that this is the line described by the set.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by two number lines: a horizontal line (x-axis) and a vertical line (y-axis). These axes intersect at a point called the origin, which has coordinates (0, 0). The plane allows us to locate and plot points using pairs of numbers known as coordinates.
- The horizontal line is the x-axis. It reflects the x-coordinate of a point.
- The vertical line is the y-axis. It reflects the y-coordinate of a point.
- Points are represented as (x, y).
X-Coordinate
The x-coordinate is the first number in an ordered pair \(x, y\) that specifies the position of a point on the coordinate plane. It indicates the horizontal placement of a point in relation to the y-axis.
- If the x-coordinate is positive, the point is to the right of the y-axis.
- If the x-coordinate is negative, the point is to the left of the y-axis.
- An x-coordinate of zero means the point lies on the y-axis.
Y-Coordinate
The y-coordinate of a point \(x, y\) measures its vertical distance from the x-axis on the coordinate plane. This value is crucial for determining the height of any point or shape in relation to the x-axis.
- Positive y-coordinates indicate a position above the x-axis.
- Negative y-coordinates indicate a position below the x-axis.
- A y-coordinate of zero means the point is on the x-axis.
Vertical Line
A vertical line is a straight line running parallel to the y-axis on the coordinate plane. It is characterized by an equation of the form x = c, where c is a constant. This equation indicates that all points on the line have the same x-coordinate, while the y-coordinate can vary.
In the exercise, the line defined by x = 3 represents a vertical line where each point has an x-coordinate equal to 3, resulting in an infinite line across all y-values.
- The line does not cross the x-axis except at its point x = 3.
- It extends upwards and downwards without any limits in y-values.
- Vertical lines do not have a defined slope.
