Question

which is the graph of $y = \frac{2}{x + 1} - 6$?

Explanation:

Step1: Analyze the function type

The function \( y = \frac{2}{x + 1}-6 \) is a rational function, which is a transformation of the parent function \( y=\frac{1}{x} \).

Step2: Find the vertical asymptote

For a rational function \( y=\frac{f(x)}{g(x)} \), the vertical asymptote occurs where \( g(x) = 0 \). Here, \( x+1=0\Rightarrow x=- 1 \), so the vertical asymptote is \( x=-1 \).

Step3: Find the horizontal asymptote

For a rational function of the form \( y=\frac{a}{x - h}+k \), the horizontal asymptote is \( y = k \). In our function, \( k=-6 \), so the horizontal asymptote is \( y=-6 \).

Step4: Analyze the graph's features

The graph should have a vertical asymptote at \( x = - 1 \) and a horizontal asymptote at \( y=-6 \). The given graph (the blue curve) has a vertical asymptote (where the graph approaches but never touches, around \( x=-1 \) - looking at the x - axis scale, the vertical asymptote is at \( x=-1 \)) and the horizontal asymptote should be \( y = - 6 \). The graph shown has parts that approach a horizontal line (the horizontal asymptote) and a vertical line (vertical asymptote) consistent with the function \( y=\frac{2}{x + 1}-6 \). Also, we can check a point. Let's take \( x = 0 \), then \( y=\frac{2}{0 + 1}-6=2 - 6=-4 \). So the graph should pass through \( (0,-4) \). Looking at the graph, when \( x = 0 \), the y - value is around - 4 (consistent with our calculation).

Answer:

The graph shown (the blue - curved graph with vertical asymptote near \( x=-1 \) and horizontal asymptote near \( y = - 6 \), passing through \( (0,-4) \)) is the graph of \( y=\frac{2}{x + 1}-6 \).