Question

identify the graph of this rational function. y = 5/(x - 1)

Explanation:

Step1: Find the vertical asymptote

For a rational function \(y=\frac{f(x)}{g(x)}\), the vertical asymptote is found by setting \(g(x) = 0\).
For \(y=\frac{5}{x - 1}\), set \(x-1=0\), so \(x = 1\).

Step2: Analyze the general form of \(y=\frac{a}{x - h}\)

The parent function \(y=\frac{1}{x}\) has a vertical asymptote \(x = 0\). The function \(y=\frac{5}{x - 1}\) is a transformation of the parent function \(y=\frac{1}{x}\). It is a horizontal shift of the parent function \(y=\frac{1}{x}\) to the right by 1 unit (because of \(x-1\) in the denominator) and a vertical stretch by a factor of 5 (because of the 5 in the numerator).

Answer:

The graph with a vertical asymptote at \(x = 1\) (the third graph in the given options) is the graph of the function \(y=\frac{5}{x - 1}\).