Question 834884
Look for vertical and horizontal asymptotes first.

Since *[tex \Large \lim_{x \to \infty} \frac{x^2 - 4}{x^2 + 8x + 15} = 1], the horizontal asymptote (as x approaches infinity) will be at x = 1. Same thing happens when x goes to negative infinity.

Set the denominator equal to zero to find possible vertical asymptotes. The equation *[tex \Large x^2 + 8x + 15] gives x = -3, -5 as roots. The numerator is nonzero in both cases, so those are your vertical asymptotes.

To graph, just note the asymptotes and whether the function approaches +/- infinity as x goes to -3 or -5 in either direction. Note that x = 1 is the horizontal asymptote.