Question

quiz: hyperbolas
hs: algebra 2b m (sequential) / 5:conic sections

1. which graph of a hyperbola represents the equation $\frac{x^2}{9} - \frac{y^2}{4} = 1$

Explanation:

Step1: Identify the standard form

The given equation is \(\frac{x^{2}}{9}-\frac{y^{2}}{4} = 1\), which is in the standard form of a hyperbola that opens horizontally: \(\frac{(x - h)^{2}}{a^{2}}-\frac{(y - k)^{2}}{b^{2}}=1\), where \((h,k)\) is the center, \(a\) is the distance from the center to the vertices along the x - axis, and \(b\) is the distance from the center to the co - vertices along the y - axis. Here, \(h = 0\), \(k = 0\) (center at the origin), \(a^{2}=9\) so \(a = 3\), and \(b^{2}=4\) so \(b = 2\).

Step2: Determine the direction of opening

Since the positive term is associated with the \(x^{2}\) term, the hyperbola opens to the left and right (horizontally). The vertices are at \((\pm a,0)=(\pm3,0)\) and the co - vertices are at \((0,\pm b)=(0,\pm2)\). To identify the correct graph, we look for a hyperbola centered at the origin, opening left and right, with vertices 3 units from the center along the x - axis and co - vertices 2 units from the center along the y - axis.

(Note: Since the actual graphs are not shown here, but based on the equation, the graph should have two branches, one opening to the right (with vertex at (3,0)) and one opening to the left (with vertex at (- 3,0)), and the asymptotes of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1\) are given by the equations \(y=\pm\frac{b}{a}x=\pm\frac{2}{3}x\))

Answer:

The graph of the hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\) is a hyperbola centered at the origin, opening horizontally (left and right), with vertices at \((\pm3,0)\) and co - vertices at \((0,\pm2)\), and asymptotes \(y = \pm\frac{2}{3}x\). (To choose the correct graph from a set of options, select the one that matches these characteristics: centered at (0,0), opening left and right, vertices 3 units from the center on the x - axis, co - vertices 2 units from the center on the y - axis)