Question

find an equation for the hyperbola described. graph the equation. center at (0,0), focus at (0,3), vertex at (0,1). an equation of the hyperbola is $\frac{y^{2}}{1}-\frac{x^{2}}{8}=1$. (use integers or fractions for any numbers in the expression.) choose the correct graph below. a. b. c. d.

Explanation:

Step1: Identify the form of hyperbola equation

Since the center is \((0,0)\), focus is \((0,3)\) and vertex is \((0,1)\), the hyperbola has a vertical transverse axis. The standard form is \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1\).

Step2: Determine the value of \(a\)

The distance from the center \((0,0)\) to the vertex \((0,1)\) is \(a\). So \(a = 1\), and \(a^{2}=1\).

Step3: Determine the value of \(c\)

The distance from the center \((0,0)\) to the focus \((0,3)\) is \(c\). So \(c = 3\).

Step4: Calculate \(b^{2}\) using the relationship \(c^{2}=a^{2}+b^{2}\)

Substitute \(a = 1\) and \(c = 3\) into \(c^{2}=a^{2}+b^{2}\).

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Step5: Analyze the graph

For the hyperbola \(\frac{y^{2}}{1}-\frac{x^{2}}{8}=1\), the vertices are at \((0,\pm1)\). Graphs A and B have vertices on the \(x -\)axis (incorrect). Graph C has center - like a pair of straight lines through the origin (not a hyperbola). Graph D has vertices on the \(y -\)axis at \((0,\pm1)\) which is consistent with the equation \(\frac{y^{2}}{1}-\frac{x^{2}}{8}=1\).

Answer:

D.