Given
The equation of the hyperbola is:
[tex]\frac{y^2}{9}\text{ -x}^2\text{ = 1}[/tex]
We can see that this is a vertical hyperbola since y is positive.
The general equation of a vertical hyperbola is:
[tex]\begin{gathered} \frac{(y-k)^2}{a^2}\text{ -}\frac{(x\text{ -h})^2}{b^2}=\text{ 1} \\ Where\text{ }(h,k)\text{ is the center} \end{gathered}[/tex]
The steps to graph a hyperbola are:
1. Determine if it is horizontal or vertical. Find the center point, a, and b.
2. Graph the center point.
3. Use the a value to find the two vertices.
4. Use the b value to draw the guiding box and asymptotes.
5. Draw the hyperbola.
Step 1: This hyperbola is vertical
center point = (0,0)
Step 2: The values of a and b
[tex]\begin{gathered} a^2=\text{ 9} \\ a\text{ = 3} \\ b^2\text{ = 1} \\ b\text{ = 1} \end{gathered}[/tex]
Step 3: Draw the guiding box:
Step 4: Draw the asymptotes
The asymptotes are diagonal lines through the corners of the box
Step 5: Finally, we draw in our hyperbola. Each half starts at the vertex and continues towards the asymptotes but never actually reaches them.
Step 6:
The center point, guiding box, and asymptotes are not technically part of the answer, so a clean version of the graph would look like this:
The graph of the hyperbola is shown below:
