Question

which of the following is the function represented by the graph? consider the graph below. options: $y = \frac{1}{(x + 3)} - 5$, $y = \frac{1}{(x - 3)} + 5$, $y = \frac{1}{(x + 5)} - 3$, $y = \frac{1}{(x - 5)} + 3$

Explanation:

Step1: Analyze the vertical asymptote

The vertical asymptote of the function \(y = \frac{1}{x}\) is \(x = 0\). For a function of the form \(y=\frac{1}{x - h}+k\), the vertical asymptote is \(x = h\). Looking at the graph, the vertical asymptote is \(x = 5\). So, in the denominator, we have \(x-5\).

Step2: Analyze the horizontal asymptote

The horizontal asymptote of the function \(y=\frac{1}{x}\) is \(y = 0\). For a function of the form \(y=\frac{1}{x - h}+k\), the horizontal asymptote is \(y=k\). Looking at the graph, the horizontal asymptote is \(y = 3\).

Answer:

\(y=\frac{1}{(x - 5)}+3\) (the fourth option)