Expert: Paul Klarreich - 3/15/2008

Question
hi  

how can i find the graph & the equation for these two statements:

 - An odd-degree function with 3 relative maximum, 3 relative minimum and negative end behavior on the right.    

- An even-degree function with 4 relative maximum, 3 relative minimum.

Answer
Questioner:   lina
Category:  Advanced Math
Private:  No
 
Subject:  polynomials
Question:  hi  

how can i find the graph & the equation for these two statements:

- An odd-degree function with 3 relative maximum, 3 relative minimum and negative end behavior on the right.    

- An even-degree function with 4 relative maximum, 3 relative minimum.
=====================================
Hi, Lina,

It would be nice to get a few more facts, but I'll see what I can do.  I can't draw it for you without those, but there are some conclusions you can draw -- math is detective work, right?  That's what makes it so exciting.

- An odd-degree function

AHA! Its left and right end behavior will be opposites.

with 3 relative maximums, 3 relative minimums

SO!  Each turning point adds 1 to the degree:

No turning points: degree (probably) 1.
Six turning points:  degree at least 7.  So it starts as

P(x) =  ax^7 or higher plus more terms (lots more).

and negative end behavior on the right.

OK,THEN!  positive e-b on the left, and  'a' is negative.

Now you can draw a nice curve (see your art teacher about that) going:

Down to a minimum.
Up to a maximum.
Down to a minimum.
Up to a maximum.
Down to a minimum.
Up to a maximum.
Down, down, down forever after that.
........................................

- An even-degree function with 4 relative maximum, 3 relative minimum.

We can shorten this now:

Degree at least 8.
Left- and right- behavior the same.
The 3 mins must be in between the 4 maxes, so you have to do:

Up to a maximum.
Down to a minimum.
Up to a maximum.
Down to a minimum.
Up to a maximum.
Down to a minimum.
Up to a maximum.
Down, down, down,....

So both end-behaviors are negative.

P(x) = - ax^8 or higher plus more terms (lots more).