Investigate the family of graphs given by the
equation
for different values of the constant a.
It is more of a challenge to try to sketch the
graphs with the ideas explored in this question
without graphing software but you may like to use
it if you have software that plots the graphs of
implicit functions or parametric functions.
Prove that the graphs are all symmetrical about
the line y=x.
Make the substitution y=tx to express the
equation in parametric form (x,y)=(f(t),g(t)).
Show that all the graphs have a maximum point
when t=21/3 and find the coordinates of the
maximum point.
Show that:
(a ) for −∞ < t < −1 the graphs lies in the
fourth quadrant with x positive and y
negative
(b ) for −1 < t < 0 the graphs lie in the second
quadrant with x negative and y positive
(c ) for 0 < t < 1 the graphs lie in the first
quadrant with 0 < y < x
(d ) for 1 < t < +∞ the graphs lie in the first
quadrant with 0 < x < y.
What happens to the graphs for t=−1?