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Functions
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Rational
Functions
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Rational
functions
- a ratio of two polynomials |
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Reciprocal function |
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Translation of the reciprocal function, called
linear rational
function |
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Basic
properties of rational functions - vertical,
horizontal and oblique or slant asymptotes
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A
line whose distance from a curve decreases to zero as the
distance from the origin increases without the limit is called
the asymptote.
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The
definition actually requires that an asymptote be the tangent to
the curve at infinity. Thus, the asymptote is a line that the
curve approaches but does not cross.
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functions that most likely have vertical, horizontal and/or
slant asymptotes are rational functions. |
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So,
vertical asymptotes occur when the
denominator of the simplified rational function is equal to 0.
Note that the simplified rational function has cancelled all
factors common to both the numerator and denominator.
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The
existence of the horizontal asymptote
is related to the degrees of both polynomials in the numerator
and the denominator of the given rational function.
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Horizontal
asymptotes occur when either, the degree of the numerator is
less then or equal to the degree of the denominator.
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the case when the degree (n)
of the numerator is less then the degree (m)
of the denominator, the x-axis
y = 0
is the asymptote. |
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If
the degrees of both polynomials, in the numerator and the
denominator, are equal then, y
= an/bm
is the horizontal asymptote, written as the ratio of their
highest degree term coefficients respectively.
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When
the degree of the numerator of a rational function is greater
than the degree of the denominator, the function has no
horizontal asymptote.
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When
the degree of the numerator of a rational function is greater
than the degree of the denominator, the function has no
horizontal asymptote.
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A
rational function will
have a slant (oblique) asymptote
if the
degree (n)
of the numerator is exactly one more than the degree (m)
of
the denominator that is if n
= m + 1.
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The
graph of a rational function will never cross its vertical
asymptote, but may cross its
horizontal or slant asymptote.
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The
graph of the reciprocal
function, equilateral or rectangular hyperbola
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The
graph of the reciprocal function y
= 1/x or y
= k/x is a
rectangular (or right) hyperbola of which
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coordinate axes. |
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If
k
> 0 then,
the function is decreasing from zero to negative
infinity and from positive infinity to zero,
i.e., the graph of the rectangular hyperbola has
two branches, in the first and third quadrants as is
shown in the right figure.
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hyperbola has two axes of
symmetry. |
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vertices, |
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Translation
of the reciprocal function, linear rational function
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Example:
Given
the rational function |
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sketch
its graph. |
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| Solution:
The
vertical and the horizontal asymptote of the linear rational
function |
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| and the
coefficient |
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the original function |
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x-intercept
at a point (x,
0), |
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y-intercept
at a point (0,
y), |
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Oblique or slant
asymptote
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The
line y =
mx
+ c is a
slant or oblique asymptote of
a function f
if
f
(x)
approaches
the line
as
x approaches infinity
(or negative infinity).
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A
rational function will
have a slant (oblique) asymptote
if the
degree (n)
of the numerator is exactly one more than the degree (m)
of
the denominator that is if n
= m + 1.
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Dividing
the two polynomials
that form a rational function,
of which the
degree
of the numerator
pn (x)
is exactly one more than the degree
of
the denominator qm
(x), then
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(x)
= Q (x) · qm (x) + R
=>
pn (x)
/ qm (x)
= Q (x) + R / qm (x) |
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where,
Q (x)
=
ax + b
is the quotient and R/qm
(x)
is the remainder with constant R.
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The
quotient Q
(x)
=
ax + b
represents the equation of the slant asymptote.
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As
x
approaches
infinity (or negative infinity),
the remainder R/qm
(x)
vanishes (tends to zero).
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Thus,
to find the equation of the slant asymptote, perform the long
division and discard the remainder.
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The
graph of a rational function will never cross its vertical
asymptote, but may cross its
horizontal or slant asymptote.
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Example:
Given
the rational function |
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sketch
its graph. |
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Solution:
The
vertical asymptote can be found by finding the root of the
denominator,
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| x + 2 = 0
=>
x = -2
is the vertical asymptote.
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Since
the
degree
of the numerator is exactly one more than the degree of
the denominator the given rational function has the slant
asymptote.
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Dividing the
numerator by the denominator
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the slant asymptote y
= x |
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and
the
remainder
3/(x + 2) that vanishes as x
approaches
positive or negative infinity.
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Contents
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Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved.
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