Consider the following curves and observe that each of them is having some special properties, called symmetry with respect to a point, with respect to a line.

**Symmetry and
Asymptotes**

** **

Consider
the following curves and observe that each of them is having some special
properties, called symmetry with respect to a point, with respect to a line.

We now
formally define the symmetry as follows :

If an
image or a curve is a mirror reflection of another image with respect to a line,
we say the image or the curve is symmetric with respect to that line. The line
is called the line of symmetry.

A curve
is said to have a *Î¸*
angle rotational symmetry with respect to a point if the curve is unchanged by
a rotation of an angle *Î¸*
with respect to that point.

A curve
may be symmetric with respect to many lines. Specifically, we consider the
symmetry with respect to the co-ordinate axes and symmetric with respect to the
origin. Mathematically, a curve *f *(* x*,*
y*)* *=* *0* *is said to be

**â€¢ Symmetric with respect to the y-axis **if

**â€¢ Symmetric with respect to the x-axis **if

**â€¢ Symmetric with respect to the
origin **if** ***f*** **(**
***x*,** ***y*** **)** **=** ***f*** **(âˆ’*x*** **,** **âˆ’** ***y*** **)** **âˆ€*x*** **,** ***y*** **or if** **(** ***x*** **,**
***y*)** **is a point on** **the graph
of the curve then so is (âˆ’ *x*
, âˆ’
*y*) . That is the curve is unchanged
if we rotate it by 180Â° about the origin.

For
instance, the curves mentioned above *x*
=
*y*^{2} , *y* = *x*^{2} and
*y* = *x*
are symmetric with respect to *x*-axis,* y*-axis and origin respectively.

** **

An
asymptote for the curve *y* =
*f* ( *x*) is a straight line which is a tangent at âˆž
to the curve. In other words the distance between the curve and the straight
line tends to zero when the points on the curve approach infinity. There are
three types of asymptotes. They are

**1. Horizontal asymptote**, which is parallel to the** ***x*** **-axis. The line** ***y*** **=** ***L*** **is said to be a horizontal asymptote for the curve* y *= *f
(x)* if either lim_{xâ†’+âˆž} *f
(x)* = L or lim_{xâ†’âˆ’âˆž} *f (x)*
= L .

**2. Vertical asymptote,** which is parallel to the* y *-axis. The line* x *= a is said to be vertical asymptote for the curve* y *= *f
(x)* if lim_{ xâ†’aâˆ’} *f (x)* = Â±âˆž or lim_{ xâ†’a+} *f (x)* =Â±âˆž.

**3. Slant asymptote**, A slant (oblique) asymptote occurs
when the polynomial in the numerator is** **a
higher degree than the polynomial in the denominator.

To find
the slant asymptote you must divide the numerator by the denominator using
either long division or synthetic division.

** **

**Example 7.66**

Find the
asymptotes of the function *f* (*x*) = 1/*x*.

**Solution**

Hence, the
required vertical asymptote is or the *y*-axis.

As the
curve is symmetric with respect to both the axes, *y* = 0 or the *x* -axis
is also an asymptote. Hence this (rectangular hyperbola) curve has both the
vertical and horizontal asymptotes.

** **

Find the
slant (oblique) asymptote for the function *f
(x)* = *x*^{2}-6*x*+7 / *x*+5.

Since
the polynomial in the numerator is a higher degree (2nd) than the denominator
(1st), we know we have a slant asymptote. To find it, we must divide the numerator
by the denominator. We can use long division to do that:

Notice
that we don't need to finish the long division problem to find the remainder.
We only need the terms that will make up the equation of the line. The slant
asymptote is *y* =
*x* âˆ’11.

As you
can see in this graph of the function, the curve approaches the slant asymptote
*y *=* x *âˆ’11
but never crosses it:

** **

Find the
asymptotes of the curve

**Solution**

Therefore,
*y* = 2 is a horizontal asymptote. This
can also be obtained by synthetic division.

Tags : Applications of Differential Calculus | Mathematics , 12th Maths : UNIT 7 : Applications of Differential Calculus

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12th Maths : UNIT 7 : Applications of Differential Calculus : Symmetry and Asymptotes | Applications of Differential Calculus | Mathematics

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