Problem 58 - College Algebra

Question

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{y^{2}}{9}-\frac{x^{2}}{1}=1\)

Step-by-Step Solution

Answer

The similarity between the graphs of these two equations is that they both have the same semi-axes lengths, which are 1 and 3. The difference is that the hyperbola defined by \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) opens horizontally, while that defined by \(\frac{y^{2}}{9}-\frac{x^{2}}{1}=1\) opens vertically.

Step 1: Identify the orientation of the first equation

For the equation \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\), the 'x' term comes first and is positive, meaning the hyperbola opens along the x-axis (horizontally). The difference in the denominators of the x and y terms signifies the length of the semi-axes a and b, being √9 = 3 and √1=1, respectively.

Step 2: Identify the orientation of the second equation

In the equation \(\frac{y^{2}}{9}-\frac{x^{2}}{1}=1\), however, the 'y' term comes first and is positive. This means that the hyperbola created from this equation will open along the y-axis (vertically). The lengths of the semi-axes would still be √1 = 1 and √9 =3, respectively.

Step 3: Formulate the Similarity and Difference

The similarity between these two hyperbolas is that they both have the same length for their semi-axes, which are 1 and 3. The difference, however, is their orientation. The first hyperbola is oriented horizontally while the second one is oriented vertically.