- Identified vertical asymptotes of \(f(x) = \frac{-2x^2 + 14}{x^2 - 49}\) at \(x = 7\) and \(x = -7\).
- Analyzed limits near asymptotes by determining signs of factors.
- Evaluated behavior: \(x \to 7^+ \implies f(x) \to -\infty\), \(x \to 7^- \implies f(x) \to \infty\), \(x \to -7^+ \implies f(x) \to \infty\), \(x \to -7^- \implies f(x) \to -\infty\).
- Determined the correct statement is: \(\boxed{\text{As } x \text{ approaches } 7 \text{ from the right, } y \text{ approaches } -\infty}\).
Explanation
- Identify Asymptotes and Factor the Function
The given function is \(f(x) = \frac{-2x^2 + 14}{x^2 - 49}\). To analyze the behavior of the graph at the vertical asymptotes, we first identify the asymptotes by finding where the denominator is zero: \(x^2 - 49 = 0 \implies (x - 7)(x + 7) = 0\). Thus, the vertical asymptotes are at \(x = 7\) and \(x = -7\). We can rewrite the function in factored form as \(f(x) = \frac{-2(x^2 - 7)}{(x - 7)(x + 7)}\).
- Analyze Behavior Near Asymptotes
We analyze the sign of \(f(x)\) as \(x\) approaches the vertical asymptotes.
At \(x = 7\):
- As \(x \to 7^-\), \(x-7\) is a small negative number, \(x+7 \approx 14\) (positive), and the numerator \(-2(x^2-7) \approx -2(49-7) = -84\) (negative). Thus, \(f(x) \approx \frac{\text{negative}}{\text{negative} \times \text{positive}} = \frac{\text{negative}}{\text{negative}} = +\infty\).
- As \(x \to 7^+\), \(x-7\) is a small positive number, \(x+7 \approx 14\) (positive), and the numerator \(-2(x^2-7)\) is negative. Thus, \(f(x) \approx \frac{\text{negative}}{\text{positive} \times \text{positive}} = \frac{\text{negative}}{\text{positive}} = -\infty\).
At \(x = -7\):
- As \(x \to -7^-\), \(x+7\) is a small negative number, \(x-7 \approx -14\) (negative), and the numerator \(-2(x^2-7)\) is negative. Thus, \(f(x) \approx \frac{\text{negative}}{\text{negative} \times \text{negative}} = \frac{\text{negative}}{\text{positive}} = -\infty\).
- As \(x \to -7^+\), \(x+7\) is a small positive number, \(x-7 \approx -14\) (negative), and the numerator \(-2(x^2-7)\) is negative. Thus, \(f(x) \approx \frac{\text{negative}}{\text{negative} \times \text{positive}} = \frac{\text{negative}}{\text{negative}} = +\infty\).
- Evaluate Choices and Conclude
Comparing our findings with the given choices:
- "As \(x\) approaches \(-7\) from the left, \(y\) approaches \(\infty\)": Incorrect, it approaches \(-\infty\).
- "As \(x\) approaches \(-7\) from the right, \(y\) approaches \(-\infty\)": Incorrect, it approaches \(\infty\).
- "As \(x\) approaches \(7\) from the left, \(y\) approaches \(-\infty\)": Incorrect, it approaches \(\infty\).
- "As \(x\) approaches \(7\) from the right, \(y\) approaches \(-\infty\)": Correct.
Therefore, the statement "As \(x\) approaches \(7\) from the right, \(y\) approaches \(-\infty\)" is the correct one.
Examples
Understanding the behavior of a function near its asymptotes is crucial in fields like physics and engineering, where certain conditions might cause variables to grow indefinitely (the "asymptote" effect). For example, if you are designing a cooling system and the efficiency function reaches an asymptote at a specific temperature, knowing whether the efficiency drops to negative infinity (indicating a total failure or breakdown) or rises to positive infinity (perhaps suggesting an unrealistic model) helps engineers set safety limits and design robust systems. Analyzing these mathematical limits allows for the prediction of system behavior near critical operational thresholds.