Hyperbola Transformation
Transformation of Rectangular Hyperbolas
The basic hyperbola is .
The transformations that we need to consider are:
- Vertical Dilation, a stretch of the graph in the direction of the y-axis
- Vertical Translation, moving the graph up and down
- Horizontal Translation, moving the graph right and left
The equation is . We are going to look at the effects of changing , , and .
is shown as a green curve. If , , and , the blue curve, will lie on top of the green curve.
Changing the value of b.
Keeping and , increase the value of .
Click on the box to show the vertical asymptote. For the original the vertical asymptote is the y-axis (). What happens to the vertical asymptote as we increase the value of ?
What happens to the vertical asymptote if we decrease the value of ?
Which transformation is ?
Changing the value of c.
Keep and return to zero.
Increase the value of . Click on the box to show the horizontal asymptote. For the original the horizontal asymptote is the x-axis (). What happens to the horizontal asymptote as we increase the value of ?
What happens to the horizontal asymptote as we decrease the value of ?
Which transformation is ?
Changing the value of a.
Start by increasing the value of , staying with positive values. What happens?
Now try some negative values, what do you notice?
Which transformation is ?
There is a good way of determining for a given graph. Click on the "Show Vector" box. Change the value of , you can change and as well. See if you can complete the sentence, "If we go one unit to the right from the point where the asymptotes cross, the value of is ...".