Now that we have graphed a hyperbola, lets do the reverse and write an equation of a hyperbola given different pieces of information.

First, if we are given the x and y-intercepts we just have to think in reverse of graphing. Remember we took the square root of the numbers under the x and y variable to place on the x and y axis. So if we are given these numbers, we just square them and place them under the appropriate variable. But we must also know which is the major axis so we know whether to put the x in front of the subtraction sign or the y. Lets say for this example, x is the major axis and we are give x-intercepts of 3, and -3 and the ends of the minor axis are at 5 and -5. Then we have the following equation:

Now lets look what we do when we are given the foci and the "difference of the focal radii". Remember that the variable c is associated with the foci and the difference of the focal radii is the length of the major axis which is always equal to 2a. So if we are given the foci of : (3,0) and (-3,0) and the difference of the focal radii is 4, then we can say that c= 3 and and 2a = 4, making a = 2.

Which equation do you think we substitute this information into now?

We take these two numbers, square them and put them in the appropriate places in the equation: b² = c² - a^2, so that now we have:
b² = 9 - 4, giving us the value of b =√5. Since the foci are on the x axis, this makes the x axis the major axis, so in the equation we are going to write, x² goes before the subtraction sign and a² goes under it, leaving y² to go after the subtraction sign and b² to go under y².

Take all this information to write the final answer. Try it and then check your answer.

Substituting all this into the final equation we now get the following equation for the final answer: