To identify the asymptotes of the hyperbola given by the equation:
\[ \frac{x^2}{16} - \frac{(y - 4)^2}{7} = 1 \]
we start by recognizing this as the standard form of a hyperbola centered at \((h, k) = (0, 4)\) with:
- \(a^2 = 16\) (thus \(a = 4\))
- \(b^2 = 7\) (thus \(b = \sqrt{7}\))
For hyperbolas of the form \(\frac{x^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\), the equations of the asymptotes are given by:
\[ y - k = \pm \frac{b}{a}(x - h) \]
Substituting in our values, we have:
- \(h = 0\)
- \(k = 4\)
- \(b = \sqrt{7}\)
- \(a = 4\)
The slopes of the asymptotes will be:
\[ \frac{b}{a} = \frac{\sqrt{7}}{4} \]
Thus, the equations of the asymptotes are:
\[ y - 4 = \pm \frac{\sqrt{7}}{4}(x - 0) \]
This simplifies to:
\[ y - 4 = \frac{\sqrt{7}}{4}x \quad \text{and} \quad y - 4 = -\frac{\sqrt{7}}{4}x \]
Therefore, we can write the asymptotes as:
\[ y = \frac{\sqrt{7}}{4}x + 4 \quad \text{and} \quad y = -\frac{\sqrt{7}}{4}x + 4 \]
From the options given, the correct response for the asymptotes is:
\[ y = \frac{\sqrt{7}}{4}(x) + 4 \quad \text{and} \quad y = -\frac{\sqrt{7}}{4}(x) + 4 \]
Thus, the response "y equals Start Fraction Start Root 7 End Root over 4 End Fraction left-parenthesis x plus 4 right-parenthesis and" would be correct if it noted that the terms within parentheses correctly modify "x" lastly.
So, the final answer for the equations of the asymptotes is:
\[ y = \frac{\sqrt{7}}{4}(x) + 4 \quad \text{and} \quad y = -\frac{\sqrt{7}}{4}(x) + 4 \]