By Carly Coffman
Letís start our exploration of hyperbolas by first exploring h and k.† Since we have explored these variables in the previous lessons, you will have a link with a given equation where you have to substitute values for h and k.† Remember you can also use the ďnĒ button on the bottom of the window to explore the variables.† Have fun!†† Exploration 1
Open a Microsoft Word document, title it ďHyperbolasĒ and type your name and date under the title.† Answer all questions using complete sentences on your Word document.
1) In the equations, †and , what do h and k represent? (You can copy and paste each equation into your Word document)
Now, letís explore a and b.† In the following exploration, click on the play button at the bottom to change the values for a.† †††† †††† Exploration 2
2) As a increases, what happens to the graph of the hyperbola?
Letís explore b now, by clicking on the play button at the bottom of the next window.
3) What affect does b have on the graph of the hyperbola?
Letís look into these values a bit further.† Look at the following hyperbola graphs.
The line from (-2,0) to (2,0) is called the transverse axis since it connects the two vertices (-2,0) and (2,0).† Notice that from the center (0,0) to each vertex there is a distance of 2.† The line from (3,0) to (-3,0) is called the conjugate axis.† Notice that from the center to (3,0) there is a distance of 3.
4)†† So, how do these distances on each axis relate to a^2 and b^2?
The asymptotes of the hyperbola (which are shown in brown) are formed by creating the diagonals of the rectangle formed by the transverse and conjugate axis. The equations of the asymptotes depend on the orientation of the transverse axis.†
ō If the transverse axis is horizontal, the asymptotes are
y = (b/a)x††† and††† y = -(b/a)x
ō If the transverse axis is vertical, the asymptotes are
y = (a/b)x††† and††† y = -(a/b)x
5) So, what are the asymptotes for the equation †(graph
6) As a increases, what happens to the graph of a hyperbola with a horizontal transverse axis? (Press the play button to view a^2 increasing.)† †Exploration 4 †
7) †† What happens to the graph of a hyperbola with a vertical transverse axis when a
†increases?††† Exploration 5
Now, letís put all of the information we have learned together.
For each of the following equations find the center of the graph and the equations of the asymptotes.
Print your Word document and place it in your portfolio or notebook with our other investigations.† Congratulations, you are finished with the hyperbola lesson!