Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d). $\text{ƒ}\left(x\right)=-1/(x-2)$

Verified step by step guidance

1. Identify the vertical asymptote by setting the denominator equal to zero: solve the equation \(x - 2 = 0\) to find \(x = 2\). This means the graph has a vertical asymptote at \(x = 2\).
2. Analyze the behavior of the function \(ƒ(x) = -\frac{1}{x-2}\) as \(x\) approaches 2 from the left side (\(x \to 2^-\)). Since \(x - 2\) is slightly negative, the denominator approaches zero through negative values, making the fraction approach positive or negative infinity depending on the sign.
3. Analyze the behavior of the function as \(x\) approaches 2 from the right side (\(x \to 2^+\)). Here, \(x - 2\) is slightly positive, so the denominator approaches zero through positive values, affecting the sign and direction of the function's values near the asymptote.
4. Summarize the behavior near the vertical asymptote: determine whether the function goes to \(+\infty\) or \(-\infty\) on each side of \(x=2\) by considering the negative sign in the numerator and the sign of the denominator on each side.
5. Match the observed behavior with one of the four given graph choices (A–D) by comparing how the graph approaches the vertical line \(x=2\) on both sides, focusing on whether the function values go to positive or negative infinity.

Key Concepts

Vertical Asymptotes

A vertical asymptote occurs where a function's denominator is zero and the function grows without bound near that x-value. For rational functions, vertical asymptotes indicate values excluded from the domain, such as x=2 in ƒ(x) = -1/(x-2). Understanding their behavior helps analyze the graph near these lines.

Behavior Near Vertical Asymptotes

The graph of a rational function near a vertical asymptote can approach infinity or negative infinity from either side. The sign and degree of the function determine whether the graph approaches the asymptote from above or below, and whether it does so symmetrically or differently on each side.

Graphing Rational Functions

Graphing rational functions involves identifying asymptotes, intercepts, and end behavior. For ƒ(x) = -1/(x-2), the vertical asymptote at x=2 and the negative sign affect the shape, causing the graph to approach the asymptote differently on each side, which is key to matching the function to its graph.