f(x)=(-2x² + 14)/(x² - 49) which statement describes the behavior of the graph of the function shown at the…

f(x)=(-2x² + 14)/(x² - 49) which statement describes the behavior of the graph of the function shown at the vertical asymptotes? o as x approaches -7 from the left, y approaches ∞. o as x approaches -7 from the right, y approaches -∞. o as x approaches 7 from the left, y approaches -∞. o as x approaches 7 from the right, y approaches -∞.

f(x)=(-2x² + 14)/(x² - 49) which statement describes the behavior of the graph of the function shown at the vertical asymptotes? o as x approaches -7 from the left, y approaches ∞. o as x approaches -7 from the right, y approaches -∞. o as x approaches 7 from the left, y approaches -∞. o as x approaches 7 from the right, y approaches -∞.

Answer

Explanation:

Step1: Find vertical asymptotes

Set the denominator $x^{2}-49 = 0$. Factoring gives $(x + 7)(x - 7)=0$. So the vertical asymptotes are $x=-7$ and $x = 7$.

Step2: Analyze behavior near $x=-7$

For $x$ near $-7$, consider the sign of the function $f(x)=\frac{-2x^{2}+14}{x^{2}-49}$. When $x\to - 7^{-}$, the numerator $-2x^{2}+14$ is negative and the denominator $(x + 7)(x - 7)$ is negative (since $x+7\to0^{-}$ and $x - 7\to - 14$), so $y\to+\infty$. When $x\to - 7^{+}$, the numerator is negative and the denominator is positive (since $x + 7\to0^{+}$ and $x - 7\to - 14$), so $y\to-\infty$.

Step3: Analyze behavior near $x = 7$

When $x\to7^{-}$, the numerator $-2x^{2}+14$ is negative and the denominator $(x + 7)(x - 7)$ is negative (since $x + 7\to14$ and $x - 7\to0^{-}$), so $y\to+\infty$. When $x\to7^{+}$, the numerator is negative and the denominator is positive (since $x + 7\to14$ and $x - 7\to0^{+}$), so $y\to-\infty$.

Answer:

As x approaches -7 from the right, y approaches -∞.