Problem 110 - College Algebra

Question

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\)

Step-by-Step Solution

Answer

At x = -2, there is a vertical asymptote. As the graph nears -2 from the left side, it shoots upward towards infinity. As the graph nears -2 from the right side, it plunges downward into negative infinity.

Step 1: Understanding the concept of vertical asymptotes

A vertical asymptote represents a value \(x = a\) that the function approaches but never quite reaches. A function behaves differently when approaching a vertical asymptote from either side of the number line. It will either steadily increase or decrease without taking on the value of the asymptote.

Step 2: Understanding the behavior of the function as x approaches -2 from the left

If \(f(x) \rightarrow \infty\) as \(x \rightarrow -2^{-}\), it means that as x values get closer and closer to -2 from the left side (or lesser side), the y-values of the function climb higher and higher. The function \(f(x)\) tends to positive infinity. This implies the graph shoots upward as it gets close to -2 from the left.

Step 3: Understanding the behavior of the function as x approaches -2 from the right

If \(f(x) \rightarrow -\infty\) as \(x \rightarrow -2^{+}\), it means that as x values get closer and closer to -2 from the right side (or greater side), the y-values of the function plummet down or become very large negative values. The function \(f(x)\) tends to negative infinity. This implies as the graph moves towards -2 from the right, it plunges downward into negative infinity.

Step 4: Analyzing and describing the overall behaviour of the graph near its vertical asymptote

Having understood the trends on both sides, it can be concluded that at x = -2, the graph has a vertical asymptote. As we approach this vertical asymptote from the left, the graph shoots upward and as we approach it from the right, the graph plunges downward.