anonymous
  • anonymous
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.
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Michele_Laino
  • 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
  • anonymous
Ok. Know what do I do?

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Michele_Laino
  • 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
  • anonymous
No. That was the whole question. That's why I was confused :(
Michele_Laino
  • 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
  • Michele_Laino
So a way to establish the x-intercept, is to record at which month, the water level W(x)=0
anonymous
  • anonymous
Is that the answer?
Michele_Laino
  • 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
  • anonymous
what about the last part? Should we create an imaginary problem with a degree higher than 3?
Michele_Laino
  • 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
  • Michele_Laino
which*
anonymous
  • anonymous
ahhh I understand! Ok so thats the final answer, correct?
Michele_Laino
  • 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
  • anonymous
Thank-you:)
Michele_Laino
  • Michele_Laino
:)
Michele_Laino
  • Michele_Laino
ok!
Michele_Laino
  • 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
  • anonymous
I agree
Michele_Laino
  • Michele_Laino
here is the corresponding graph:
Michele_Laino
  • 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
  • Michele_Laino
yes! I think so!
Michele_Laino
  • 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
  • anonymous
I honestly wanna say thank-you for all of your help::)
Michele_Laino
  • Michele_Laino
:)
anonymous
  • anonymous
Sorry, I have one last question!
Michele_Laino
  • Michele_Laino
I'm pondering...
Michele_Laino
  • 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
  • Michele_Laino
so, Tucker and Karly are both correct, if they refer to a graph like this: |dw:1436024150337:dw|
Michele_Laino
  • Michele_Laino
|dw:1436024243604:dw|
Michele_Laino
  • Michele_Laino
|dw:1436024282359:dw|
anonymous
  • anonymous
OHHH So how could we write that into an equation for the both of them?
Michele_Laino
  • Michele_Laino
|dw:1436024389509:dw|
Michele_Laino
  • Michele_Laino
yes!
anonymous
  • 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
  • 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
  • anonymous
Ohh :) Thank-you once again!!!!
Michele_Laino
  • Michele_Laino
:)
anonymous
  • anonymous
Are you up for one more?
Michele_Laino
  • Michele_Laino
ok!
Michele_Laino
  • 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
  • anonymous
ok
Michele_Laino
  • 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
  • anonymous
yeas. Would it be x3-2x2-9x+18?
Michele_Laino
  • 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
  • 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
  • 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
  • anonymous
i see, I see. Now what do we do?
Michele_Laino
  • 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
  • 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
  • anonymous
Is that it?
Michele_Laino
  • Michele_Laino
yes! it is the second part, we have to answer to the first part
anonymous
  • anonymous
wouldn't we factor x3-2x2-9x+18?
Michele_Laino
  • Michele_Laino
yes! I have factored that polynomial
Michele_Laino
  • Michele_Laino
\[{x^3} - 2{x^2} - 9x + 18 = \left( {x - 2} \right)\left( {x + 3} \right)\left( {x - 3} \right)\]
anonymous
  • anonymous
so would it be (x-2)(x+3)(x-3) as an answer?
anonymous
  • anonymous
oh okay now what do we do?
Michele_Laino
  • 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
  • anonymous
ok, I understand
anonymous
  • anonymous
now we have to answer the 2nd part yes?
Michele_Laino
  • Michele_Laino
we have to say the graph type which represents our original polynomial: \[{x^4} - {x^3} - 11{x^2} + 9x + 18\]
anonymous
  • anonymous
ok.
anonymous
  • anonymous
So what would the graph look like?
Michele_Laino
  • Michele_Laino
here is my reasoning:
Michele_Laino
  • Michele_Laino
let's consider the point x=2 for exmple
anonymous
  • anonymous
ok
Michele_Laino
  • 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
  • anonymous
ohhh I see. How would we graph w/o technology?
Michele_Laino
  • 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
  • Michele_Laino
necessarily it is a s follows: |dw:1436026592697:dw|
anonymous
  • anonymous
:) Thank-you so much you're a lifesaver!!!!
anonymous
  • anonymous
how long will u be on here? I may need help later
Michele_Laino
  • Michele_Laino
or like this: |dw:1436026652043:dw|
Michele_Laino
  • Michele_Laino
I will stay here in Open Study, for at least 1 hour
anonymous
  • anonymous
ok:) Thanks I will email you again soon
Michele_Laino
  • Michele_Laino
:)

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