Problem 11 - Algebra 2

Question

Describe the vertical asymptotes and holes for the graph of each rational function. $$ y=\frac{x+5}{x+5} $$

Step-by-Step Solution

Answer

The graph of the function \(y = \frac{x+5}{x+5}\) has a hole at x = -5 and no vertical asymptotes.

Step 1: Setting the denominator equal to zero

To find the vertical asymptotes, set the denominator equal to zero and solve for x. This gives: \[x + 5 = 0\] Solving for \(x\) yields \(x = -5\).

Step 2: Checking for common factors

Next, check whether any common factors can be cancelled out in the given expression. In the given equation,there is a common factor in the numerator and the denominator (\(x + 5\)), which can be cancelled out. When these cancel out, this indicates there is a hole in the graph, not a vertical asymptote. This is because the common factor exists in both the numerator and the denominator.

Step 3: Conclusion

From the above steps, there are no vertical asymptotes since the common factor \(x + 5\) is cancelled out. So, the graph will have a hole at x = -5.

Key Concepts

Understanding Vertical Asymptotes

Vertical asymptotes are vital features in the analysis of rational functions. They occur where the denominator of the rational function equals zero, leading to a division by zero, which is undefined. In simple terms, it's where the y-values of the graph head towards infinity as they approach the vertical line.
When dealing with a rational function, such as \( y = \frac{f(x)}{g(x)} \), finding vertical asymptotes involves setting the denominator \( g(x) \) equal to zero and solving for \( x \).
However, just because a factor causes the denominator to be zero does not always mean we have a vertical asymptote, as illustrated in our original example. In the function \( y = \frac{x+5}{x+5} \), we initially identify the potential for a vertical asymptote at \( x = -5 \) by setting \( x + 5 = 0 \). But, remember to verify if the factor appears in both the numerator and the denominator because that changes things.

Identifying Holes in Graphs

Holes in the graph of a rational function occur when both the numerator and the denominator share a common factor, which cancels out leaving a hole. The hole represents a point of discontinuity in the function, which means the function is not defined at a specific point, even though the limit as \( x \) approaches this point exists.
In the earlier example \( y = \frac{x+5}{x+5} \), the common factor \( x + 5 \) appears in both the numerator and the denominator. When this factor is cancelled, not only do we not have a vertical asymptote but instead we have a hole at \( x = -5 \). Remember that the graph of the function looks the same as \( y = 1 \) except for this particular discontinuity.
Key points to remember about holes:

• They appear when a factor is cancelled out completely from both the numerator and denominator.
• They cause the function to be undefined at that specific point.
• The graph will exhibit behavior close to linear, but with a break at the hole location.

Simplifying Rational Expressions

Simplifying rational expressions is a straightforward yet essential step in understanding their behavior and characteristics, such as identifying asymptotes and holes.
To simplify, we factor both the numerator and denominator to identify any common factors. When common factors exist, we cancel them out. This simplification often reveals the true nature of the graph.
For example, in \( y = \frac{x+5}{x+5} \), simplification is achieved by cancelling out \( x+5 \) from both the numerator and the denominator, which means the simplified function is simply \( y = 1 \) over the domain except where there's a hole at \( x = -5 \).
Always remember:

• Simplifying helps us more easily find holes since any factor that cancels out reveals a hole, not an asymptote.
• It assists in sketching a rough graphical approximation by clarifying the form of the function.

Thus, simplification is not just a mathematical exercise but a pathway to more deeply understanding the function's graph.