Final answer:
The graph of the function f(x) = -2x² + 14 / (x² - 49) approaches negative infinity at its vertical asymptotes which are x = ±7.
Step-by-step explanation:
The behavior of the graph of the function f(x) = -2x² + 14 / (x² - 49) at the vertical asymptotes can be described by examining the denominator of the function, x² - 49. The vertical asymptotes occur where the denominator equals zero, which in this case, are at x = ±7. To determine how the graph behaves around these asymptotes, we analyze the sign of the function as x approaches these critical values. Since the numerator is always negative (as it is a negative quadratic), the sign of the function as x approaches the asymptotes from the left or the right will depend on the sign of the denominator.
A vertical asymptote's behavior indicates that, for rational functions, the graph will approach infinity or negative infinity as the value of x gets closer to the x-coordinate of the asymptote. In this specific case, for x-values near the vertical asymptotes, specifically just before -7 and just after 7, the function f(x) approaches negative infinity because the numerator is negative and the denominator is positive. Thus, the correct statement describes the behavior as option B) The graph approaches negative infinity.