Polynomials and rational functions are smaller groups of Algebraic Functions
Another group of Algebraic Functions are Rational Power
Functions.
A rational power function is a function where the exponent
is not an integer.
f ( x)  x
m
n
m
is the rational power
n
n is an integer greater than1
m and n have no common factors.
Rational functions can be written different ways:
x  x   x
m
n
n
m
n
m
When the rational powers are positive the graph increases.
When the rational powers are negative the graph decreases.
If m is greater than n, the graph will approach infinity quickly.
If m is less than n, the graph will approach infinity slowly.
y

y






x
m<n





m>n
x
Ex 1: Sketch the following graphs:
a.) f ( x)  2 x  2  1
Solution:
b.) f ( x)  (4 x  3)
3
2
a.) this graph will be shifted 2 units left and 1
unit down.
The y-coordinates will be multiplied by a
factor of 2. (vertical elongation)
The parent function is the square root
function.
y
Since m<n, the graph will increase slowly.
2 





 -1
x
b.)
The graph will be shifted ¾ of a unit to the right.
f ( x)  4x 
3
4

3
2
3
2
3
3 2
4
3
3 2
4
 4 ( x  )  8( x  )
y
All of the output is multiplied
by 8 (vertical elongation).




x
The graph is not a line!
Ex 2: Sketch the graph of
Solution:
x2
0
x2
f ( x) 
x2
x2
Use a sign graph to find the
domain and to see how the
graph approaches the vertical
asymptote.
(x - 2) - - - - - - - - - - - - - - - 0 + + + D f  (,2)  [2, )
(x + 2) - - - - 0 + + + + + + + + + + + As x  2  , f ( x)  
fcn
+++ --------- 0+++
vert asymptote:
x = -2
________________________
Horiz. Asymptote:
-2
0
2
x-int: x = 2
y-int: none


 
y=1
y


x
x2
Ex 3: Sketch the graph of f ( x) 
x2  x  2
Notice that the root is in the denominator of the rational function.
When this occurs, you will have two horizontal asymptotes.
Recall: when you take the square root of a value you get two
solutions: one positive and one negative.
f ( x) 
x2
x2

( x  2)( x  1)
x2  x  2
x ≠ -2, 1
(x + 2) - - - 0 + + + + + + + + + + +
(x – 1) - - - - - - - - - - - 0 + + + + + +
fcn
++
------ 0++++++
_________________________
-2 -1 0 1 2
Denominator cannot
be zero. The radicand
cannot be zero nor
negative.
D f  (,2)  [1, )
vert. asmyptotes: x = -2, 1
x-int: x = 2
y-int: none
The horizontal asymptotes are the tricky ones!!! 
x2
x2
f ( x) 

1
2
2
x  x  2 ( x  x  2) 2
f ( x) 
x2
( x  x  2)
2
1
2

x2
21
2
Now distribute the power!
We are only concerned with the
first term here.
x ...
x2
x2
f ( x)  21 
x 2 ... x...
Horizontal asymptotes: y = ±1
Since the degrees of both
polynomials is 1, we have
horizontal asymptotes at the
ratio of their leading coefficients.
We have two because the denominator
of this function was a square root.
y
 




(x – 2) - - - - - - - - - - - - - - 0 + + + +
(x + 2) - - - 0 + + + + + + + + + + +
(x – 1) - - - - - - - - - - - 0 + + + + + +
fcn
--- +++++ 0-0++++
_________________________
-2 -1 0 1 2
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x