1.What is the upper bound of the function?

f(x)=4x^{4}−25x^{2}−5x−13

Enter your answer in the box.

upper bound =

2.

What is the end behavior of the polynomial function?

Drag the choices into the underlines to correctly describe the end behavior of the function. The option for both could be either ** f(x)→−∞ or f(x)→∞**

** f(x)=−7x ^{3}−x+1 **

As x→−∞

As x→−∞

** f(x)=2x ^{4}+6x^{2}+4**

As x→∞

As x→∞

jjennylove Sep 10, 2019

#1**+1 **

1.What is the upper bound of the function?

f(x)=4x^4−25x^2−5x−13

I'll admit....I had this in Pre-Cal ages ago, but I forgot how to calculate the upper and lower bounds on the zeroes without a littlte "refresher"....here are the steps to finding the upper [ and lower] bounds on the zeroes:

1. The leading coefficient must be a "1".....if not.....divide all the terms by the leading coefficient.....so we have

x^4 - (25/4)x^2 - (5/4) - (13/4)

2. Write down all the coefficients → 1, (-25/4) , (-5/4) , (-13/4)

3. "Throw away" the lead coefficient , 1

4. Remove the minus signs ....so...we are left with the values 25/4, 5/4 , 13/4

5. We now have two possible bounds :

(a) Bound 1 = the largest value + 1 = (25/4) + 1 = (25/4) + (4/4) = 29/4

(b) Bound 2 = the sum of all values or 1, whichever is larger.....so...

25/4 + 5/4 + 13/4 = 43/4 which is larger than 1

The smallest of these two possible bounds is our answer = 29/4 = 7.25

This tells us that that the upper and lower values on zeroes are -7.25 and 7.25

So....the upper bound is 7.25

Look at the graph here: https://www.desmos.com/calculator/wdwslnqzqa

The function has two real roots.....notice that they are contained between x = -7.25 and x =7.25

I do not know why this works....I expect that the proof of it is difficult !!!!

BTW : here is a good site that reviews this : https://www.mathsisfun.com/algebra/polynomials-bounds-zeros.html

CPhill Sep 10, 2019

#3**0 **

wouldnt than answer be just 13 since whn you do long divsion 13/4 -25 -5 -13 it is 4 27 346 and 4485 which are all postitive ?

jjennylove
Sep 12, 2019

#2**+1 **

Second one

f(x) = -7x^3 - x + 1

Remember the rule .....if the lead coefficient is negative and the power on its associated variable is odd:

The poynomial will tend toward + infinity as x → - infinity

And....the polynomial will tend tword -infinity as x → infinity

See the graph here : https://www.desmos.com/calculator/y47dnl4gy5

f(x) = 2x^4 + 6x^2 + 4

The lead coefficient is positive and the power on its associated variable is even

The polynomial will approach + infinity as x → -infinity and as x → infinity

See the graph here : https://www.desmos.com/calculator/lmgxbyiyzo

CPhill Sep 10, 2019

#4**+1 **

thanks for your help! I found out that the answer for the last one was actaully postitive infinty as well. Just ated to let you know so you can make note of it.Thanks again though for your help i aprreacite it.

f(x) = 2x^4 + 6x^2 + 4

The lead coefficient is positive and the power on its associated variable is even

The polynomial will approach + infinity as x → **infinity ** and as x → infinity

jjennylove
Sep 12, 2019