Wednesday, June 3, 2009

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)

14 comments:

Kate Nowak said...

Nice start - I recommend you spend some more time thinking about what people really need to know, succinctly, instead of trying to transcribe your notes.

Kate Nowak said...

Also, let me compliment you on your formatting. This is very easy to read.

J.D. Fisher said...

Great graphs!

It always surprises me to see discontinuous (not sure if that's the right word) graphs made from plain ol' real numbers.

Reminds me of heartbeat monitors. Gotta be some connection there, I would think.

Kate Nowak said...

Lovely job, Anna and Rachel.

I like how when Anna is Really Serious, she prefaces the comment with "Just for the record..."

:-)

Ellen + Kelsey said...

hey guys justr wanted to say that your blog is rly easy to read and i like the format!

Anna said...

Thanks everyone! I hope it helps

kelseyyyy said...

Hey guys! This really helped me study the rational functions. It was really easy to read and much better than trying to go back through my notes. THANKS!!

Nicole S. said...

Wow guys this is awsome you rock! it really sums everything up well.. i just have one question tho.. on the third example question, if the graph has a multiplicity of three, how come the numerator is -x and not x^3?

Anna said...

Hi Nicole. I was a little confused when I put it down too. But we know it has to have VAs of -3 and 3, so the denom is x^2-9. But it also has to have an HA of y=0. In order for the HA to be y=0 the numerator's exponent must be less than the denom's. Hence the x. As for the negative, as far as I know, you just have to fool around with it in on your calc. Hope I helped!

julia said...

you guys basically just saved my life. thank you for helping me not fail the final!

ps. this looks too professional, you should charge us to use it

Kate Nowak said...

See that, Julia? Blogging SAVED A LIFE.

Steve said...

Ya, this is a nice post. Lots of the stuff I forgot was from this unit :p. I completely forgot how to find asymptotes until this post.

BFa said...

steeeeveee

BFa said...

that was really helpful, thanks