S70 TRACING OF CURVES. 



Also when x = + a we see that y is infinite. 

 Hence x a, 



and x = a 



are asymptotes. 



We may now assign different values to x, and note the 



d *u 

 corresponding values of y and -f- obtained from (1) and (2). 



CLX 



Since the curve is symmetrical with respect to the axis of x, 

 we may confine our attention to the positive values of y. 



dy 



When x = 0, y = 0, -- = 2. 



From x = to x = a, y is possible. 

 When x = a, y = oo , -3" = oc . 



From x = a to x = 2a, y is impossible. 



When a;=2a, v = 0, -/=. 



ax 



When x is greater than 2a, y is possible. 



It is not necessary to give negative values to x in this 

 example, because the curve is symmetrical with respect to 

 the axis of y. 



If we draw the asymptotes and make use of the above 



list of particular values of y and -&- , we shall have sufficient 



dsc 



materials for ascertaining the general form of the curve. If 

 nece'ssaryj in any example, we may find -~ t , in order to 

 determine the points of inflexion ; also by examining when ~ 



vanishes, we can determine the maxima and minima values 

 ofy. 



If we take the upper sign in equation (3), we have for 

 the asymptote 



y = x (4) ; 



A f ^ 3a2 P 



and tor the curve v = x - &c (5). 



