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

Explanation:

Step1: Identify Vertical Asymptote

The function is ( y = \frac{2}{x + 1}-6 ). The vertical asymptote occurs where the denominator is zero: ( x + 1 = 0 \Rightarrow x=-1 ). So the vertical asymptote is ( x = -1 ).

Step2: Identify Horizontal Asymptote

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

Step3: Analyze Graph Behavior

• The vertical asymptote ( x = -1 ) (a vertical line at ( x=-1 )) and horizontal asymptote ( y=-6 ) (a horizontal line at ( y = -6 )) should guide us.
• The first graph (left - most) has vertical asymptote near ( x=-1 ) (since the grid has ( x=-5,-10,-15 ) and the curve approaches a vertical line around ( x=-1 ) when considering the grid? Wait, no, wait the first graph: let's check the horizontal asymptote. The first graph has horizontal asymptote around ( y=-6 ) (since the horizontal line is around ( y=-6 )) and vertical asymptote ( x=-1 ). Wait, the three graphs:

First graph: vertical asymptote at ( x=-1 ) (the red curves approach a vertical line near ( x=-1 )), horizontal asymptote ( y=-6 ) (the horizontal line the curves approach is ( y=-6 )).

Second graph: vertical asymptote at ( x=-5 ) (denominator ( x + 5 ) maybe?), horizontal asymptote ( y = 0 )? No, our function has horizontal asymptote ( y=-6 ).

Third graph: vertical asymptote at ( x = 0 ) (denominator ( x )), horizontal asymptote ( y = 0 )? No.

Wait, re - checking: the function ( y=\frac{2}{x + 1}-6 ) is a transformation of ( y=\frac{2}{x} ). We shift ( y=\frac{2}{x} ) left by 1 unit (because of ( x+1 )) and down by 6 units (because of ( -6 )). The vertical asymptote of ( y=\frac{2}{x} ) is ( x = 0 ), after shifting left 1, it's ( x=-1 ). The horizontal asymptote of ( y=\frac{2}{x} ) is ( y = 0 ), after shifting down 6, it's ( y=-6 ).

So the graph with vertical asymptote ( x=-1 ) and horizontal asymptote ( y=-6 ) is the first graph (left - most).

Answer:

The Left - most Graph (the first graph in the given three - graph set)