The basic hyperbola is [math]y=\frac{1}{x}[/math]. [br][br]The transformations that we need to consider are:[br][br] - Vertical Dilation, a stretch of the graph in the direction of the y-axis[br] - Vertical Translation, moving the graph up and down[br] - Horizontal Translation, moving the graph right and left[br][br]The equation is [math]y=\frac{a}{x-b}+c[/math]. We are going to look at the effects of changing [math]a[/math], [math]b[/math], and [math]c[/math].[br][br][math]y=\frac{1}{x}[/math] is shown as a green curve. If [math]a=1[/math], [math]b=0[/math], and [math]c=0[/math], the blue curve, [math]y=\frac{a}{x-b}+c[/math] will lie on top of the green curve.

Keeping [math]a=1[/math] and [math]c=0[/math], increase the value of [math]b[/math].[br][br]Click on the box to show the vertical asymptote. For the original [math]y=\frac{1}{x}[/math] the vertical asymptote is the y-axis ([math]x=0[/math]). What happens to the vertical asymptote as we increase the value of [math]b[/math]?[br][br]What happens to the vertical asymptote if we decrease the value of [math]b[/math]?[br][br]Which transformation is [math]b[/math]?

Keep [math]a=1[/math] and return [math]b[/math] to zero.[br][br]Increase the value of [math]c[/math]. Click on the box to show the horizontal asymptote. For the original [math]y=\frac{1}{x}[/math] the horizontal asymptote is the x-axis ([math]y=0[/math]). What happens to the horizontal asymptote as we increase the value of [math]c[/math]?[br][br]What happens to the horizontal asymptote as we decrease the value of [math]c[/math]?[br][br]Which transformation is [math]c[/math]?

Start by increasing the value of [math]a[/math], staying with positive values. What happens?[br][br]Now try some negative values, what do you notice?[br][br]Which transformation is [math]a[/math]?[br][br]There is a good way of determining [math]a[/math] for a given graph. Click on the "Show Vector" box. Change the value of [math]a[/math], you can change [math]b[/math] and [math]c[/math] 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 [math]a[/math] is ...".