Chapter 9.3, Problem 8E

To determine

To match: The graph that represents the equation $\frac{y^2}{9}-\frac{x^2}{25}=1$ from the given graphs.

  

Answer to Problem 8E

The graph that represents the equation $\frac{y^2}{9}-\frac{x^2}{25}=1$ is graph (d).

Explanation of Solution

Given information:

The hyperbola equation is $\frac{y^2}{9}-\frac{x^2}{25}=1$ . The four given graphs are as follows.

  

Calculation:

If $x$ goes first in the equation, the equation has a horizontal transverse axis, and if $y$ goes first in the equation, the equation has a vertical transverse axis.

In the hyperbola equation $\frac{y^2}{9}-\frac{x^2}{25}=1$ , $y$ comes first, so this hyperbola equation has vertical transverse axis.

Compare the equation $\frac{y^2}{9}-\frac{x^2}{25}=1$ with standard equation $\frac{{\left(y-k\right)}^2}{a^2}-\frac{{\left(x-h\right)}^2}{b^2}=1$ .

  $\begin{array}{c}a=3\\ b=5\\ \left(h,k\right)=\left(0,0\right)\end{array}$

The transverse axis is vertical. The center is at $\left(0,0\right)$ . The distance from one of the vertices to the center is 3.

Therefore, the graph that represents the equation $\frac{y^2}{9}-\frac{x^2}{25}=1$ is graph (d).