Problem 10 - Algebra 2

Question

Describe the vertical asymptotes and holes for the graph of each rational function. $$ y=\frac{3}{x+2} $$

Step-by-Step Solution

Answer

The function \(y=\frac{3}{x+2}\) has a vertical asymptote at \(x = -2\) and does not have any holes.

Step 1: Identify the undefined points of the function

To find the points where the function is undefined, set the denominator equal to zero and solve for \(x\). So, calculate \(x+2 = 0\) which results in \(x = -2\). Therefore, \(x = -2\) is the undefined point of the function.

Step 2: Identify if there is any factor common to both numerator and denominator

If a factor is common to both the numerator and denominator, it would indicate the presence of a hole. In this case, the numerator is a constant (3) and the denominator is \(x+2\). These expressions have no common factors that would cancel out, so no holes are present.

Step 3: Find the vertical asymptote

As there is no hole, the undefined point \(x = -2\) is a vertical asymptote of the function. A vertical asymptote represents a value that the function approaches but never reaches. In graphical terms, it is a vertical line that the graph approaches asymptotically as \(x\) approaches a certain value.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are key features of rational functions, and they appear where the function tends to infinity. When you have a rational function like \( y=\frac{3}{x+2} \), the first step to finding vertical asymptotes is to determine where the function is undefined.

• To find these points, set the denominator to zero and solve for \( x \).
• In this example, solving \( x+2=0 \) gives you \( x=-2 \).

This result indicates a vertical asymptote at \( x=-2 \). In graphical terms, the graph of the function will approach this vertical line but never actually cross or touch it. It's important to note that vertical asymptotes represent a barrier that the graph cannot breach. This helps in sketching the graph accurately as it clarifies how the function behaves as it nears certain \( x \)-values.

Holes in Graphs

Holes in the graph of a rational function occur when a factor is canceled out between the numerator and the denominator. They result in undefined points, but unlike vertical asymptotes, the function does not tend to infinity at these points. Instead, there is simply a "gap" or hole. In the rational function \( y=\frac{3}{x+2} \), the numerator and denominator don't share any common factors, as 3 is a constant and cannot cancel with \( x+2 \).

• Since no factors are canceled, there are no holes in this function's graph.

Understanding the presence or absence of holes is crucial, as they affect graph behavior differently than asymptotes. Holes don’t cause the function to shoot off to infinity but rather leave you with a disconnected point on the graph.

Undefined Points

Undefined points in a function occur when the denominator of a rational function equals zero, but these points can manifest as either vertical asymptotes or holes, depending on the factorization. In the given function \( y=\frac{3}{x+2} \), solving \( x+2=0 \) indicates that the function is undefined at \( x=-2 \).

• If no common factors are present between numerator and denominator, the undefined point results in a vertical asymptote.
• If a common factor exists, it cancels out, often resulting in a hole instead.

Here, there are no common factors, so the point is a vertical asymptote. Recognizing undefined points helps mark significant moments in graph behavior and is essential for predicting where the graph can have gaps or infinite behavior.