Question

find the center, transverse axis, vertices, foci, and asymptotes. graph the equation.

the center is at (0,0)
(type an ordered pair. type exact answers for each coordinate, using radicals as needed.)

the transverse axis is along the x - axis.

the vertices are at
(simplify your answer. type ordered pairs. type exact answers, using radicals as needed. use a comma answers as needed.)

Explanation:

Step1: Recall the standard form of hyperbola

The standard form of a hyperbola is \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\), where \((h,k)\) is the center. For the given equation \(\frac{x^2}{16}-\frac{y^2}{9}=1\), we have \(h = 0,k = 0,a^2=16,a = 4,b^2 = 9,b=3\).

Step2: Find the vertices

The vertices of a hyperbola \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\) are \((h\pm a,k)\). Substituting \(h = 0,k = 0,a = 4\), we get \((0\pm4,0)\).

Answer:

\((- 4,0),(4,0)\)