Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Graph each line. Also, give the (a) \(x\) -intercept (if any), (b) \(y\) -intercept (if anyy, (c) domain, (d) range, and (e) slope of the line (if defined). $$x=2$$

Step-by-Step Solution

Answer

Vertical line, \(x\)-intercept at \(x=2\), no \(y\)-intercept, domain \(\{2\}\), range \((-\infty, \infty)\), undefined slope.

Step 1: Identify the type of line

The equation given is \(x = 2\). This is a vertical line because it describes a line that is parallel to the y-axis and crosses the x-axis at \(x = 2\).

Step 2: Determine the intercepts

For a vertical line \(x = 2\), there is an \(x\)-intercept at \(x = 2\). Since the line runs parallel to the y-axis, there is no \(y\)-intercept because it never crosses the y-axis.

Step 3: Find the domain and range

The domain of a vertical line \(x = 2\) is just \(\{2\}\) because \(x\) is always 2. The range, however, is all real numbers, \((-\infty, \infty)\), because \(y\) can be any value.

Step 4: Determine the slope

A vertical line has an undefined slope, because it rises vertically without running horizontally. Hence, the slope for the line \(x = 2\) is undefined.

Key Concepts

Vertical Line

A vertical line is a straight line that runs up and down on a graph and is parallel to the y-axis. In the equation form, such as \(x = 2\), the line shows that for every point on this line, the \(x\) value is constant. This means that the line passes through the x-axis exactly at \(x = 2\) and never touches the y-axis. Because it does not shift left or right, each point on this line has the \(x\) value of 2.

It’s quite unique compared to regular linear equations like \(y = mx + b\) where both \(x\) and \(y\) can change across the line. Instead, with a vertical line:

• All points have the same x-coordinate.
• The y-coordinate can take any possible value, which makes its range all real numbers.

This characteristic makes vertical lines especially easy to spot once you realize they simply go straight up and down and remain fixed at a specific \(x\)-value.

Domain and Range

When graphing lines, understanding the domain and range is crucial. The domain of a line refers to all possible \(x\)-values that the line can have, whereas the range refers to all possible \(y\)-values the line can cover.

For a vertical line like \(x = 2\), the domain is incredibly straightforward. It doesn't stretch across a range of \(x\) values. Instead, its domain is limited to exactly one value: \(x = 2\). Thus, the domain is written as \(\{2\}\), indicating the single \(x\)-value the line can have.

In contrast, the range of a vertical line is unrestricted. Since the line goes up and down without stopping, \(y\) can be any number. Therefore, the range of \(x = 2\) is all real numbers, noted as \((-\infty, \infty)\). This highlights how flexible the y-coordinate is on a vertical line.

Undefined Slope

The concept of slope is essential in understanding linear equations. Slope refers to how steep a line is, expressed as the ratio of the change in \(y\) (rise) over the change in \(x\) (run). For most lines, you can calculate it easily. However, a vertical line is special.

For the line \(x = 2\), you notice there is no run. The x-value stays the same, which means there's no horizontal movement. Since you are practically dividing by zero (something mathematically impossible) when trying to find a slope for a vertical line, the slope is termed as "undefined".

Vertical lines like \(x = 2\) don't have conventional slopes, but understanding they have an undefined slope helps highlight their unique nature. Here are some key takeaways:

• The line rises infinitely without any run.
• There is no rate of change in the x-direction.

Remembering that vertical lines have undefined slopes can help in graphing and analyzing lines correctly.