The change in water level of a lake is modeled by a polynomial function, W(x). Describe how to find the x-intercepts of W(x) and how to construct a rough graph of W(x) so that the Parks Department can predict when there will be no change in the water level. You may create a sample polynomial of degree 3 or higher to use in your explanations.

- anonymous

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- anonymous

@Michele_Laino Ok

- Michele_Laino

a general formula for a polynomial of third degree can be this:
\[\Large W\left( x \right) = A{x^3} + B{x^2} + Cx + D\]
where A, B, C, and D are real coefficients

- anonymous

Ok. Know what do I do?

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## More answers

- Michele_Laino

in order to determine those coefficients, we need to know some data about the water level of your lake.
Do you have a table which collects the water level of the lake in function of time?

- anonymous

No. That was the whole question. That's why I was confused :(

- Michele_Laino

if we draw a graph in which time x is along the horizontal axis, and the water level W(x) is along the vertical axis, then the x-intercept, is the month at which W(x)=0

- Michele_Laino

So a way to establish the x-intercept, is to record at which month, the water level W(x)=0

- anonymous

Is that the answer?

- Michele_Laino

the complete answer should report the values of each coefficient A, B, C, and D, and the followed procedure to get those values. Nevertheless, without any data about the water level of the lake, we are not able to evaluate those coefficients

- anonymous

what about the last part? Should we create an imaginary problem with a degree higher than 3?

- Michele_Laino

Using a polynomial whose degree is greater than 3, involves many real coefficients whixh have to be determined. For example, if we conjecture a polynomial of degree 4, then we can write:
\[\Large W\left( x \right) = {A_1}{x^4} + {A_2}{x^3} + {A_3}{x^2} + {A_4}x + {A_5}\]
As you can see, now we have to determine, by using our experimental observations, 5 real coefficients, namely:
\[\Large {A_1},\;{A_2},\;{A_3},\;{A_4},\;{A_5}\]

- Michele_Laino

which*

- anonymous

ahhh I understand! Ok so thats the final answer, correct?

- Michele_Laino

yes! You can write this:
"We can establish the values of each coefficients of our sample polynomial function for W(x), by experimental observations about the water level W(x) as function of time or as function of the months of the year"

- anonymous

Thank-you:)

- Michele_Laino

:)

- Michele_Laino

ok!

- Michele_Laino

I think that better is if we make a drawing of that function:
\[\Large T\left( x \right) = {\left( {x - 4} \right)^3} + 6\]
That function is represented by a cubic parabola

- anonymous

I agree

- Michele_Laino

here is the corresponding graph:

##### 1 Attachment

- Michele_Laino

as we can see the turning point is at x=4.
At x=4 the temperature is:
\[\Large T\left( 4 \right) = {\left( {4 - 4} \right)^3} + 6 = 0 + 6 = 6\]

- Michele_Laino

yes! I think so!

- Michele_Laino

The requested experimental procedure, which can be used in order to find that turning point, can be this:
"We substitute many values for the x variable, when a change in x, produces little change in T(x), then we are close to that turning point"

- anonymous

I honestly wanna say thank-you for all of your help::)

- Michele_Laino

:)

- anonymous

Sorry, I have one last question!

- Michele_Laino

I'm pondering...

- Michele_Laino

as stated in the previous exercise, a turning point can be like a vertex of a parabola, or a quadratic function

- Michele_Laino

so, Tucker and Karly are both correct, if they refer to a graph like this:
|dw:1436024150337:dw|

- Michele_Laino

|dw:1436024243604:dw|

- Michele_Laino

|dw:1436024282359:dw|

- anonymous

OHHH So how could we write that into an equation for the both of them?

- Michele_Laino

|dw:1436024389509:dw|

- Michele_Laino

yes!

- anonymous

i understand how to do it now but I just don't know how to put it into words, if that makes sense.

- Michele_Laino

a possible sentence is like this:
"Tucker and Karly are saying the same thing, so they can both be correct, if they refer to a graph like this, for example:"
|dw:1436024700576:dw|

- anonymous

Ohh :) Thank-you once again!!!!

- Michele_Laino

:)

- anonymous

Are you up for one more?

- Michele_Laino

ok!

- Michele_Laino

That graph is a polynomial of fourth degree:
\[\large P\left( x \right) = {x^4} - {x^3} - 11{x^2} + 9x + 18\]

- anonymous

ok

- Michele_Laino

In order to establish the point x such that P(x)=0, we have to factorize those function P(x).
Do you know how to factorize that function?

- anonymous

yeas. Would it be x3-2x2-9x+18?

- Michele_Laino

better is:
\[{x^4} - {x^3} - 11{x^2} + 9x + 18 = \left( {x + 1} \right)\left( {{x^3} - 2{x^2} - 9x + 18} \right)\]

- anonymous

Wow. I can't believe I got that right! Usually I get those equations wrong. R u sure lol? I mmay have gotten that wrong

- Michele_Laino

now, we can factorize the polynomial:
x3-2x2-9x+18, so the complete factorization of P(x) is:
\[\large {x^4} - {x^3} - 11{x^2} + 9x + 18 = \left( {x + 1} \right)\left( {x + 3} \right)\left( {x - 3} \right)\left( {x - 2} \right)\]

- anonymous

i see, I see. Now what do we do?

- Michele_Laino

now the points x, such that P(x)=0, are given by the subsequent conditions:
x+1=0 --->x=-1
x+3=0 ---> x=-3
x-3=0 --->x=3
x-2=0 ---> x=2

- Michele_Laino

now, since x is a profit, then it has to be a positive number, so our acceptable solutions are:
x=2, and x=3

- anonymous

Is that it?

- Michele_Laino

yes! it is the second part, we have to answer to the first part

- anonymous

wouldn't we factor x3-2x2-9x+18?

- Michele_Laino

yes! I have factored that polynomial

- Michele_Laino

\[{x^3} - 2{x^2} - 9x + 18 = \left( {x - 2} \right)\left( {x + 3} \right)\left( {x - 3} \right)\]

- anonymous

so would it be (x-2)(x+3)(x-3) as an answer?

- anonymous

oh okay now what do we do?

- Michele_Laino

as I wrote before, the complete factorization is:
\[\left( {x + 1} \right)\left( {x + 3} \right)\left( {x - 3} \right)\left( {x - 2} \right)\]

- anonymous

ok, I understand

- anonymous

now we have to answer the 2nd part yes?

- Michele_Laino

we have to say the graph type which represents our original polynomial:
\[{x^4} - {x^3} - 11{x^2} + 9x + 18\]

- anonymous

ok.

- anonymous

So what would the graph look like?

- Michele_Laino

here is my reasoning:

- Michele_Laino

let's consider the point x=2 for exmple

- anonymous

ok

- Michele_Laino

if we substitute x=2.1, into the equation of P(x), we get a negative quantity, since, we have:
x-2--->2.1-2 = 0.1 >0
x+1---> 2.1 +1 =3.1 >0
x-3---> 2.1-3=-0.9 <0
x+3---> 2.1+3=5.1 >0
then the product:
\[\left( {x + 1} \right)\left( {x + 3} \right)\left( {x - 3} \right)\left( {x - 2} \right)\]
is negative

- anonymous

ohhh I see. How would we graph w/o technology?

- Michele_Laino

I think that the subsequent reasoning is better:
since our graph has to pass at subsequent points:
\[\left( { - 3,0} \right),\;\left( { - 1,0} \right),\;\left( {2,0} \right),\;\left( {3,0} \right)\]

- Michele_Laino

necessarily it is a s follows:
|dw:1436026592697:dw|

- anonymous

:) Thank-you so much you're a lifesaver!!!!

- anonymous

how long will u be on here? I may need help later

- Michele_Laino

or like this:
|dw:1436026652043:dw|

- Michele_Laino

I will stay here in Open Study, for at least 1 hour

- anonymous

ok:) Thanks I will email you again soon

- Michele_Laino

:)

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