First, let's identify the key components of the equation [tex]\(\frac{(x+2)^2}{4^2} - \frac{(y-3)^2}{5^2} = 1\)[/tex]. This is the standard form of the equation for a hyperbola that opens horizontally because of the positive term corresponding to [tex]\(x\)[/tex].
1. Center of the Hyperbola: The center of the hyperbola is given by [tex]\((h, k)\)[/tex]. From the given equation, we have [tex]\(h = -2\)[/tex] and [tex]\(k = 3\)[/tex]. Hence, the center is:
[tex]\[
(-2, 3)
\][/tex]
2. Vertices of the Hyperbola: For a hyperbola that opens horizontally, the vertices are located at [tex]\((h \pm a, k)\)[/tex], where [tex]\(a\)[/tex] is the value associated with the denominator of the [tex]\(x\)[/tex]-term. Here, since [tex]\( \frac{(x+2)^2}{4^2} \)[/tex] gives us [tex]\(a = 4\)[/tex].
- The left vertex is:
[tex]\[
(h - a, k) = (-2 - 4, 3) = (-6, 3)
\][/tex]
- The right vertex is:
[tex]\[
(h + a, k) = (-2 + 4, 3) = (2, 3)
\][/tex]
Thus, the correct answers are:
- The center of the hyperbola is [tex]\((-2, 3)\)[/tex].
- The left vertex is [tex]\((-6, 3)\)[/tex].
- The right vertex (or the other vertex) is [tex]\((2, 3)\)[/tex].
