Regions of the Curve

Now let we will discuss about Regions of the Curve. 

-----  Solve the equation for  y.

If  y   is imaginary when   x   lies between   a  and   b,

then the curve does not lie in the region bounded by 

x=a   and x=b

-----  Find asymptotes parallel to axis and the curve will not go beyond its asymptotes.

-----  Sometimes it is possible to detect values of 

x   and   y    for which two sides of the equation assume opposite signs.

The curve does not exist for such values.

Increase or Decrease of the Curve:-

-----  Solve the equation for  y  or  x   whichever is found convenient.

now see the behaviour of  y   or  x   for different values of  

x  or  y  giving particular attention to those values for which 

y  or  x  tends to infinity or zero.

If there is symmetry about axis of   x   or   y   i.e. in opposite quadrant, only positive values need be considered. The other branches are drawn by symmetry.

-----  Find   (dy/dx)  and points where tangents are parallel to axis. 

There are maxima and minima (which already discussed in my previous post in this blog) of the curve and here are the ordinates cease to increase or decrease.