1. If x is replaced by -x but the equation remains the same, the graph shall be symmetrical about y-axis.

2. If y is replaced by - y but the equation remains the same, the graph shall be symmetrical about x-axis.

   For example : y² = 4ax is symmetrical about x-axis.

   x² + y² = a² is symmetrical about both x and y axis.

1. If both x and y are changed by - x and - y respectively and simultaneously,and equation is unchanged, then the graph is symmetrical about the origin and symmetrical in opposite quadrants .

   In above illustration, x² + y² = a² is a standard circle having center at the origin with radius ‘a’ units.

2. When x and y are interchanged and equation remains the same, the curve is symmetrical about the line y = x.

   Clearly the curve x³ + y³ = 3axy is symmetrical about y = x.

Intercepts on the axes : Put y = 0 in the equation to find x-intercept and put x = 0 to find y-intercept.

Extent of the curve : Know the domain of the curve that exists. This gives the horizontal extent. Know the range, this gives the vertical extent of the curve (i.e. function)

Tangent at the origin : If the curve passes through origin, then equate its lowest degree term to zero. This gives the tangents at the origin.

For example, y² = 4ax, passes through origin

\ Taking 4ax = 0 gives x = 0 ( i.e. y-axis )

Similarly x² = 4by, passes through origin

\ Taking 4by = 0 gives y = 0 ( i.e. x - axis )

The curve x³ + y³ = 3axy also passes through origin.

\ Taking 3axy = 0 or x = 0 and y = 0 i.e. both axes touch the curve at the origin.

Asymptotes : The value of x which makes y infinitely large gives a straight line which touches a branch of the curve at infinity. Similarly for y.

Such a straight line is called an Asymptote to the curve.

1. To locate vertical asymptotes (parallel to y-axis), if an equation of nth degree, the term with yn is absent, then the coefficient of the next highest power of y when equated to zero, gives the vertical asymptotes (provided this coefficient must not be a constant.

   For example, x³+ 3xy² + y² + 2x + y = 0

   Since y³ is absent. The coefficient of y² is ( 3x + 1 ) = 0 which the asymptote parallel to y-axis i.e. vertical asymptote.