Sketch of curves

Sketch of curves

Whenever we find area it is important to have rough sketch of required portion. For sketching we should keep following points in our mind

(a) Symmetry

If on replacing x by -x in the equation, equation of curve remains same means curve is symmetric about y - axis for example x2 = 4by. Also if on replacing y by -y equation of curve remains same means curve is symmetric about x - axis. For example y2 = 4ax.

(b) Origin

Whether curve passes through origin or not. If constant term in the equation is zero ⇒ Curve passes through origin.

(c) Intersection points with axes

Check on what points curve crosses x and y axes. First put y = 0 and solve for x to find intersection points with x axis and put x = 0 to find intersection points with y - axis.

(d) Asymptotes

Find out asymptotes of the curve. First find out value of y in terms of x and then if for any real value of x, y approaches +∞ or −∞ then x equal to that real value is asymptote to that curve. For example xy = c2 ⇒ y = c2/x. If x → 0, y → ∞ so x = 0 line is asymptote to the curve xy = c2. Similarly x = c2/y. Now as y → 0, x → ∞ so y = 0 is also asymptote.

(e) Standard Graphs

Graphs which you have studied in function must remember. Also maximum and minimum value of a function helps in sketching of a curve.