Rational Functions!

Rational Functions!

DEFINITION: a rational function is a polynomial divided by a polynomial

Examples:

AND
(corresponding to graphs seen below)

* Rational functions often have guidelines called ASYMPTOTES (shown above in red)

• The first graph (see above) has a vertical asymptote of x=0 and a horizontal asymptote of y=0.
• The second graph has two VAs and one HA.
  
  

To FIND vertical asymptotes: set the denominator equal to zero.

Ex: set x=0 ------ VA (x=0) ---- Domain is All Reals except 0

Ex: set to 0 ------ VA (x= +/- 2) ---- Domain: Reals except +/- 2

Ex: set to 0 ------ NO VA ---- Domain: All Reals

Algebraic Rules for horizontal asymptotes (HA) for the rational function:

1. If n is less than m, then the HA is y=0 (x-axis)
2. If n=m, then the HA is y= a/b
3. If n>m, then there is no HA

Examples:

1. ------ 1<2 y="0

2. ------ 2=2 (n=m) ---- HA: y= 2/3

3. ------ 3>2 (n>m) ---- NO HA

REVIEW: Roots, Y-intercept, x/- Intervals, Symmetry

1. Roots: set numerator equal to zero ---- ---- ---- x= +/- 1

2. Y-intercept: sub in zero ---- = = ------ (0, 1/4)

3. +/- Intervals: breakpoints (roots and discontinuities) {+/-1, +/-2}

VA: x= +/- 2

Then...
Test f(x) in each interval with a number line numbered from (in this case) -2 through 2

4. Symmetry: f(-x) = f(x) ---- Even (about y-axis)

5. VA: x= +/-2 HA: y=1

6. Sketch:

The EFFECT Of Multiplicity of Roots on Graph at VA's

is the same as

VA: ------ x=0, 4 (mult 2) *****

HA: 3=3 ---- 1/1 ---- y=1

Roots: ---- x= -2 and x=2 (mult 2)

Number line...

AND

The Actual Graph:

Common Factors in Numerator and Denominator ---- Hole in the Graph

Graph will be the same shape as except undefined at x=2

On calculator table: for x=0 and 2 --- y will show "error"

** Cancel common factor -- shape of graph -- factors that cancel are the hole -- factors that don't cancel are the VA **

Algebraic Rules For Slant Asymptotes

A function f(x) has a slant asymptote iff ("if and only if") the degree of the numerator is exactly one more than the degree of the denominator.

and n-m=1

To sum things up:

• if n is less than m --- y=0
• n=m --- y=a/b
• n-m=1 --- slant
  

The equation of the slant asymptote is the quotient of the numerator divided by the denominator (disregard any remainder)

EXAMPLE:

You do long division just as you would with normal numbers.

• This comes out with a remainder of 27x-26
• Completely disregard the remainder
• The equation on top is the equation of the slant asymptote!

In the end, you get a slant asymptote of y=2x-7

Lastly...

GIVEN GRAPH, Write The Equation

1.

Roots: (1,0)

• Numerator: (x-1)
• VA: x=2
• Denominator: (x-2)

Solution:

2.

Roots: (-1,0) (1,0)

• Num: (x+1)(x-1)=
• VA: x=2, x=-2
• Denom: (x-2)(x+2)=

Solution:

3.

Roots: (0,0) mult 3

• Num:
• VA: x=3, x= -3
• Denom: (x-3)(x+3)=

Solution:

** Sometimes you just have to fool around with it in your calculator- plug in a negative or an integer.

** Just for the record, when a graph "bounces" at a root- the muliplicity of that root is 2.

And now for...

REVIEW PROBLEMS

1. Give the equation of the horizontal asymptote of .

2. Give the equation of the slant asymptote of .

3. Find the equations of all vertical asymptotes of .

4. Write the equation of a rational function which has a horizontal asymptote of y=3/2 and y-intercept of -2.

5. Write the equation of a rational function which has roots 0 and -5 and whose vertical asymptotes are x=+/-2.

6. Given the graph, find the equation of the rational function.

7. Sketch the rational function .

8. Find the correct equation for the graph below that has a SA of y= 1/2x + 5/4 and a VA of x= 1/2

9. Given the equation, , find the horizontal and vertical asymptotes.

10. For the following equation, , find the roots, the y-intercept, vertical asymptotes, horizontal asymptote, +/- intervals, symmetry, and domain.



---------------------------------------------------- Answers below --------------------

Answers:

1. HA: y= -4

2. SA: y= 2x+4

3. VA: x= -3/2 and x=2

4.

5.

6.

7.

8.

9.

VA: x= 5/2 and x= -1

HA: y= 5

10.

Roots:+/-i (no real roots)

Y-int: (0,1)

VA: x= +/- 1

HA: y=1

Intervals: - (less than -1), + (-1-0), + (0-1), - (greater than 1)

Symmetry: y-axis (x=0)

D: (-infinity, -1)(-1,1)(1, infinity)