To analyze the graph of the function \( y = 3 \tan^{-1}(-x) + 5 \), we need to consider the behavior of the inverse tangent function.
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Behavior of \( \tan^{-1}(x) \):
- The function \( y = \tan^{-1}(x) \) has horizontal asymptotes at \( y = -\frac{\pi}{2} \) as \( x \to -\infty \) and at \( y = \frac{\pi}{2} \) as \( x \to \infty \).
- The function increases from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).
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Behavior of \( \tan^{-1}(-x) \):
- The reflection across the y-axis implies that \( \tan^{-1}(-x) \) heads towards \( \frac{\pi}{2} \) as \( x \to -\infty \) and \( -\frac{\pi}{2} \) as \( x \to \infty \).
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The transformation \( 3 \tan^{-1}(-x) + 5 \):
- The multiplication by 3 stretches the function vertically, so the asymptotes become \( 3 \cdot \left(-\frac{\pi}{2}\right) + 5 \approx 5 - 4.71 \approx 0.29 \) as \( x \to \infty \) (lower asymptote) and \( 3 \cdot \left(\frac{\pi}{2}\right) + 5 \approx 5 + 4.71 \approx 10.71 \) as \( x \to -\infty \) (upper asymptote).
- Therefore, as \( x \to -\infty \), \( y \to 10.71 \) and as \( x \to \infty \), \( y \to 0.29 \).
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Point at \( x = 0 \):
- At \( x = 0 \), \( y = 3 \tan^{-1}(0) + 5 = 3(0) + 5 = 5 \), giving us the point \( (0, 5) \).
Based on this analysis, the key characteristics of the graph should be:
- It will be increasing.
- It crosses through \( (0, 5) \).
- The behavior will show asymptotic approach at \( y \approx 10.71 \) as \( x \to -\infty \) and at \( y \approx 0.29 \) as \( x \to \infty \).
Considering the given response options, the characteristic of increasing from an upper asymptote to passing through \( (0, 5) \) to a lower asymptote best describes the following:
The first response, the graph increasing from an asymptote of about \( y = 0 \) through \( (0, 5) \) to an asymptote of about \( y = 10 \), is the correct choice.