Answer
the vertices are $(3\pm 4,-5)$
the foci are $(3\pm \sqrt{65},-5)$
the center is $(3,-5)$
asymptotes
$$ y= \frac{7}{4}x-\frac{41}{4},$$
$$ y= -\frac{7}{4}x+\frac{1}{4}$$
Work Step by Step
The equation
$$
\left(\frac{x-3}{4}\right)^{2}-\left(\frac{y+5}{7}\right)^{2}=1
$$
is a hyperbola with $a=4, b=7$ and hence $ c=\sqrt{a^2+b^2}=\sqrt{65} $. So, we have:
- the vertices are $(3\pm 4,-5)$
- the foci are $(3\pm \sqrt{65},-5)$
- the center is $(3,-5)$
- asymptotes $y+5=\pm \frac{7}{4}(x-3)$:
$$ y= \frac{7}{4}x - \frac{21}{4}-5=\frac{7}{4}x-\frac{41}{4},$$
$$ y= -\frac{7}{4}x + \frac{21}{4}-5=-\frac{7}{4}x+\frac{1}{4}$$
