4.5.2 Rational Functions and Models - Part II
SLANT (OBLIQUE) ASYMPTOTE FOR P/Q:
 if deg P > deg Q then there is no horizontal asymptote
 function shoots off-scale at extreme left and right
 but if deg P is just 1 more than the degree of Q
 it shoots off in a very special way:
o along a non-horizontal, straight line
x2  2x 1
Example: the graph of
looks like this:
x 1

y


   





x




Note:
 the vertical asymptote is x = 1, as shown
 the slant asymptote is the (slanted) line y = x + 3, shown
 notice that the graph approaches the slant asymptote at the
extreme right and left of the graph
Q: How can we determine what the slant asymptote is?
A: Divide numerator by denominator; the slant asymptote is the
quotient:
1| 1
2
1
1
3
1
3 | 4  quotient = x + 3
4.5.2-1
Rational Equations
Rational equation: an equation that has rational
expressions (P/Q-style fractions) in it. To solve equations
with fractions:
 clear of fractions (see p. R-44):
 you get the LCD (LCM – see p. R-41)
 you multiply all terms of the equation by the LCD to:
GET RID OF THOSE NASTY FRACTIONS!!! (YEAHHHH!)
IF YOU SEE THIS:
2x
(x - 2)(x + 2)
+

1
x-2

=
it’s an EQUATION!!!
2
x+2
DO NOT (I REPEAT), DO NOT, ADD FRACTIONS!!

DO, YES, DO!! CLEAR OF FRACTIONS! 
LCD:
(x - 2)(x + 2)

2x
1 
2

(x - 2)(x + 2) (x - 2)(x + 2) + x - 2 = (x - 2)(x + 2)(x + 2)


2x + (x + 2) = (x - 2)(2)
Voila!! Hoo-Ha!! No fractions!!
3x + 2 = 2x - 4  x = -6
2(-6)
1
2
Check: (-6 - 2)(-6 + 2) + -6 - 2 =? (-6 + 2)
-12
1
2
+
=?
(-8)(-4) -8
(-4) (yes, it checks!)
NOTE: checking of answers for equations with fractions is
mandatory, because of possible zero denominators!
4.5.2-2
Variation
Recall:
if y = kx, we say that y varies directly as x
New terminology:
y = kxn
y=
k
xn

y varies directly as the nth power of x
 y varies inversely as the nth power of x
4.5.2-3