10-2 Hyperbolas Day 1 Standard Equation and the Graph

The Definition The set of all points in a plane such that the difference of the distances from two points, called foci, is constant. Does that look familiar?

Lets go back to the definition: The set of all points in a plane such that the difference of the distances from two points, called foci, is constant. (h,k)

The Picture and Equation (h,k) Tranverse axis a a c c Tree Trunk

The Other Orientation (h,k) a a c c Happy/Sad

Day 1 is simply drawing the figure You need to draw the center, the tranverse axis endpoints (still a) and the asymptotes. We will use the “box method” (more later on that) to make sure that the shape is accurate. Do you remember what the relationship between a, b and c was in ellipses?

How to find foci This time, you take the sum of the denominators:

Remember: With ellipses, you move the number under each variable in that direction. It will be a very similar method with hyperbolas. Now is when we introduce the “Box Method”

The Box Method We will draw a “box” that has the slope of the asymptotes. When you draw (and extend) the diagonals of the box, you have the asymptotes which will then allow you to draw the correct shape of the hyperbola.

The Box Method 1.Move “a” units from center to TA endpoints. 2.Move “b” units  to TA and put a little x. 3.Draw a box through the 4 points. 4.Draw the asymptotes (the diagonals of the box) 5.Draw hyperbola through TA endpoints, using asymptotes to shape the hyperbola.

Examples