Polynomials and rational functions are smaller groups of Algebraic Functions

Another group of Algebraic Functions are Rational PowerFunctions.

A rational power function is a function where the exponentis not an integer.

nm

xxf )( powerrationaltheisnm

n is an integer greater than1

m and n have no common factors.

Rational functions can be written different ways:

n mmnnm

xxx

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

x

m<n

y

x

m>n

Ex 1: Sketch the following graphs:

122)(.) xxfa 23

)34()(.) xxfb

Solution: 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.

Since m<n, the graph will increase slowly.y

x

2

-1

b.) The graph will be shifted ¾ of a unit to the right.

23

432

3

432

323

43 )(8)(44)( xxxxf

y

x

All of the output is multipliedby 8 (vertical elongation).

The graph is not a line!

Ex 2: Sketch the graph of 2

2)(

x

xxf

Solution: 02

2 x

x Use a sign graph to find thedomain and to see how thegraph approaches the verticalasymptote.

(x - 2) - - - - - - - - - - - - - - - 0 + + +(x + 2) - - - - 0 + + + + + + + + + + +fcn + + + - - - - - - - - - 0 + + + ________________________ -2 0 2

),2[)2,( fD

)(,2 xfxAsvert asymptote: x = -2Horiz. Asymptote: y = 1

x-int: x = 2

y-int: none

y

x

Ex 3: Sketch the graph of 2

2)(

2 xx

xxf

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.

)1)(2(2

2

2)(

2

xxx

xx

xxf Denominator cannot

be zero. The radicandcannot be zero nor negative.

x ≠ -2, 1

(x + 2) - - - 0 + + + + + + + + + + + (x – 1) - - - - - - - - - - - 0 + + + + + +fcn + + - - - - - - 0 + + + + + + _________________________ -2 -1 0 1 2

),1[)2,( fD

vert. asmyptotes: x = -2, 1x-int: x = 2

y-int: none

The horizontal asymptotes are the tricky ones!!!

...

2

)2(

2)(

212

21

2

x

x

xx

xxf

Now distribute the power!21

22

)2(

2

2

2)(

xx

x

xx

xxf

We are only concerned with thefirst term here.

...2

...

2)(

212 x

x

x

xxf

Since the degrees of bothpolynomials is 1, we have horizontal asymptotes at theratio of their leading coefficients.

Horizontal asymptotes: y = ±1We have two because the denominatorof this function was a square root.

y

x

(x – 2) - - - - - - - - - - - - - - 0 + + + +(x + 2) - - - 0 + + + + + + + + + + + (x – 1) - - - - - - - - - - - 0 + + + + + +fcn - - - + + + + + 0 - 0 + + + + _________________________ -2 -1 0 1 2