1.3 Rules For Drawing The Graph Of A Curve (representing
a function) :
If x is replaced by -x but the equation remains
the same, the graph shall be symmetrical about y-axis.
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
x2 + y2 = a2
is symmetrical about both x and y axis.
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.
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
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
4ax = 0 gives x = 0 ( i.e. y-axis )
Similarly x2 = 4by, passes
4by = 0 gives y = 0 ( i.e. x - axis )
The curve x3 + y3
= 3axy also passes through origin.
3axy = 0 or x = 0 and y = 0 i.e. both axes touch the curve at
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.
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.
1.1 Functions And Mapping
1.2 Functions, Their Graphs and Classification
1.3 Rules for Drawing the Graph of a Curve
1.4 Classification of Functions
1.5 Standard Forms for the equation of a straight line
1.6 Circular Function and Trigonometry
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