Part 2/5:
Here, (c) is the distance from the center to the foci, calculated by the relation (c^2 = a^2 + b^2). The vertices, asymptotes, and foci form the skeleton of the hyperbola's structure.
Changing the Equation
When the order of the variables changes, resulting in the equation:
[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 ]
the graph will shift accordingly. The vertices will now be located at points (0, ±a), indicating that the hyperbola opens upwards and downwards instead of horizontally.
Example 1: Sketching a Hyperbola
Consider the equation:
[ 9x^2 - 16y^2 = 144 ]
To express it in the standard form of a hyperbola, divide through by 144:
[ \frac{x^2}{16} - \frac{y^2}{9} = 1 ]
From this, we identify (a^2 = 16) (thus (a = 4)) and (b^2 = 9) (thus (b = 3)).