Question

identify the graph of this rational function.
y = \frac{2}{3x + 6}

Explanation:

Step1: Find vertical asymptote

Set denominator to 0: $3x + 6 = 0$
Solve for $x$: $3x = -6 \implies x = -2$

Step2: Find horizontal asymptote

Degree of numerator < denominator, so $y = 0$

Step3: Test sign of $y$

For $x > -2$ (e.g., $x=0$): $y = \frac{2}{6} = \frac{1}{3} > 0$
For $x < -2$ (e.g., $x=-3$): $y = \frac{2}{-9+6} = \frac{2}{-3} < 0$
This means the graph is above the x-axis when $x > -2$, below when $x < -2$.

Answer:

The middle graph (second option, with the curve above the x-axis for $x > -2$ and below for $x < -2$, vertical asymptote at $x=-2$ and horizontal asymptote at $y=0$)