Question

graph $\frac{x^{2}}{36}-\frac{y^{2}}{9}=1$.

Explanation:

Step1: Identify the type of conic section

The equation \(\frac{x^{2}}{36}-\frac{y^{2}}{9} = 1\) is of the form \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\), which is a hyperbola centered at the origin \((0,0)\) with a horizontal transverse axis.

Step2: Find the values of \(a\) and \(b\)

For the equation \(\frac{x^{2}}{36}-\frac{y^{2}}{9} = 1\), we have \(a^{2}=36\), so \(a = 6\), and \(b^{2}=9\), so \(b = 3\).

Step3: Analyze the vertices

The vertices of a hyperbola of the form \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) are \((\pm a,0)\). Substituting \(a = 6\), the vertices are \((\pm6,0)\).

Step4: Analyze the asymptotes

The equations of the asymptotes for a hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) are \(y=\pm\frac{b}{a}x\). Substituting \(a = 6\) and \(b = 3\), the asymptotes are \(y=\pm\frac{3}{6}x=\pm\frac{1}{2}x\).

The first graph is an ellipse (incorrect as our equation is a hyperbola). The second graph is a hyperbola with vertices on the \(x -\)axis (correct as per our analysis of \(\frac{x^{2}}{36}-\frac{y^{2}}{9} = 1\)).

Answer:

The second graph (the one with a hyperbola opening left - right with vertices on the \(x -\)axis).