Activity A.1.3.
(a)
What transformation is done to convert between the circle given by \(x^2+y^2=1\) and the graph of \(\frac{x^2}{4}+y^2=1\text{?}\) You should be specific about how the graph of \(\frac{x^2}{4}+y^2=1\) is different than the graph of \(x^2+y^2=1\text{.}\)
(b)
What transformations are done to convert between the circle given by \(x^2+y^2=1\) and the graph of \(\frac{x^2}{4}+\frac{y^2}{9}=1\text{?}\) You should 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.
(c)
What transformations are done to convert between the circle given by \(x^2+y^2=1\) and the graph of \(\frac{(x+2)^2}{4}+\frac{(y-3)^2}{9}=1\text{?}\) You should be specific about how the graph of \(\frac{(x+2)^2}{4}+\frac{(y-3)^2}{9}=1\) is different than the graph of \(x^2+y^2=1\) and specify if the transformations need to be done in a particular order.
(d)
Draw a plot of \(\frac{(x+2)^2}{4}+\frac{(y-3)^2}{9}=1\) and label the center of your plot and the points that demonstrate how far the ellipse is stretched in the vertical and horizontal directions.
(e)
The graph of the equation \(9x^2+16y^2=400\) is an ellipse. 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 ellipse and label all extreme points on your plot.
(f)
The graph of the equation \(4x^2+y^2+24x-2y+21=0\) is an ellipse. 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 ellipse and label all extreme points on your plot.
(g)
Give the equation of the ellipse shown in Figure A.1.3.
