Question

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

Explanation:

Step1: Find vertical asymptotes

Set denominator $x^2 + 3 = 0$. Since $x^2 \geq 0$, $x^2 + 3 \geq 3 > 0$ for all real $x$. No vertical asymptotes.

Step2: Find horizontal asymptote

As $x \to \pm\infty$, $x^2$ dominates, so $\lim_{x\to\pm\infty} \frac{3}{x^2+3} = 0$. Horizontal asymptote: $y=0$.

Step3: Find y-intercept

Set $x=0$: $y = \frac{3}{0^2+3} = 1$.

Step4: Analyze sign of y

Numerator $3>0$, denominator $x^2+3>0$, so $y>0$ for all $x$.

Step5: Match to graphs

Only the second graph has no vertical asymptotes, positive y-values, y-intercept at 1, and approaches $y=0$ at $\pm\infty$.

Answer:

The middle graph (with no vertical asymptotes, positive y-values, peaking at (0,1) and approaching y=0 as x approaches ±5)