Problem 27 - A Graphical Approach to College Algebra

Question

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). Do not use a calculator. \(x=2\)

Step-by-Step Solution

Answer

(a) x-intercept: (2, y); (b) no y-intercept; (c) domain: \(\{2\}\); (d) range: \((-\infty, \infty)\); (e) slope: undefined.

Step 1: Identify the Equation

We are given the equation \(x = 2\). This is a vertical line.

Step 2: Determine the x-Intercept

For a vertical line like \(x = 2\), the entire line crosses the x-axis where \(x\) is equal to 2. Therefore, the x-intercept is \((2, y)\) where \(y\) can be any real number.

Step 3: Determine the y-Intercept

Vertical lines do not have a y-intercept, as they never cross the y-axis. Therefore, there is no y-intercept for \(x = 2\).

Step 4: Find the Domain

The domain of a vertical line \(x = 2\) is restricted because the x-value is constant. Thus, the domain is \( \{2\} \).

Step 5: Find the Range

The range of a vertical line is all real numbers, \(( -\infty, \infty)\), because the line extends infinitely up and down the y-axis.

Step 6: Determine the Slope

The slope of a vertical line is undefined because a vertical line has no horizontal change. Thus, there is no defined slope for \(x = 2\).

Key Concepts

x-intercept

The x-intercept is where a line crosses the x-axis. For the equation of a vertical line like \(x=2\), this is particularly straightforward. The line crosses the x-axis at the point \((2, y)\), meaning any point along the line has an x-coordinate of 2. This is because the line maintains a constant x-value but allows any y-value. As a result, we can identify the x-intercept as \((2, y)\), where \(y\) is any real number. It is important to note that this can also be represented simply as \(x = 2\), emphasizing the fixed x-coordinate.

y-intercept

A y-intercept occurs where a line crosses the y-axis. However, when dealing with a vertical line like \(x = 2\), we run into a peculiar scenario. Vertical lines do not cross the y-axis at any point—except at infinity, which is an abstract concept—not any specific point we can identify. Therefore, the equation \(x = 2\) has no y-intercept. Since the line is parallel to the y-axis, it does not meet it at any finite point, hence no intersection point exists on the y-axis.

Domain

The domain of a function or a line refers to all possible x-values that it can take. For a vertical line like \(x = 2\), the domain is very restricted. It consists of only a single value, which is the constant x-coordinate of the line. This means the domain is simply \(\{2\}\). Unlike other lines or curves that might cover a range of x-values, a vertical line remains fixed at one x-value, resulting in its domain being just \(\{2\}\).

Range

The range of a line or function contains all possible y-values that it can assume. For vertical lines, the concept of range is quite extensive. In the case of \(x = 2\), the line stretches infinitely in both the upward and downward directions. This means its range is all real numbers, represented in interval notation as \((-\infty, \infty)\). Despite the fixed x-value, there are no limitations on y-values along this line, providing an unlimited range from negative to positive infinity.

Slope

The slope of a line defines its steepness and direction and is mathematically expressed as the change in y over the change in x (\(\frac{\Delta y}{\Delta x}\)). In the case of vertical lines, such as \(x = 2\), calculating the slope becomes problematic. The line's vertical nature means there is no horizontal change; \(\Delta x\) is zero. Because division by zero is undefined in mathematics, the slope of a vertical line is likewise undefined. Vertical lines stand straight up, without the inclination associated with having a defined slope.