Active Calculus - Multivariable

The most basic equation for a hyperbola is \(x^2-y^2=1\text{.}\) Make a plot of the hyperbola given by \(x^2-y^2=1\text{.}\) In your plot, be sure to include the asymptotes, which are given by \(y=\pm x\text{,}\) as well as the vertices. These are the points on the hyperbola that are closest to the center. In this case, the vertices are the \(x\)-intercepts.

What transformation is done to change the graph of the hyperbola given by \(x^2-y^2=1\) to the graph of \(\frac{x^2}{4}-y^2=1\text{?}\) Be specific about how the graph of \(\frac{x^2}{4}-y^2=1\) is different than the graph of \(x^2-y^2=1\text{.}\)

What transformation is done to change the graph of the hyperbola given by \(x^2-y^2=1\) to the graph of \(y^2-x^2=1\text{?}\) Be specific about how the graph of \(y^2-x^2=1\) is different than the graph of \(x^2-y^2=1\text{.}\)

What transformations are done to change the graph of the hyperbola \(x^2-y^2=1\) to the graph of \(\frac{x^2}{4}-\frac{y^2}{9}=1\text{?}\) Be specific about how the graph of \(\frac{x^2}{4}-\frac{y^2}{9}=1\) is different than the graph of \(x^2-y^2=1\) and specify if the transformations need to be done in a particular order.

Draw a graph of \(\frac{(x+2)^2}{4}-\frac{(y-3)^2}{9}=1\) and label the center, the vertices, and the asymptote lines for the hyperbola. You will need to apply the transformations from the previous part to the asymptotes \(y = \pm x\) of the base hyperbola in order to get the equations of the transformed asymptotes.

The graph of the equation \(9x^2-16y^2=400\) is a hyperbola. Convert this equation to the form \(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\text{.}\) Then use the idea of transformations from above to graph this hyperbola and label the center, the vertices, and the asymptote lines of the hyperbola.

The graph of the equation \(4x^2-y^2+24x-2y+21=0\) is an hyperbola. Convert this equation to the form \(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\text{.}\) Then use the idea of transformations from above to graph this hyperbola and label the center, the vertices, and the asymptote lines of the hyperbola.

Give the equation of the hyperbola shown in Figure A.1.6