Question

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.

Answer 1

@Michele_Laino Ok

Answer 2

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

Answer 3

Ok. Know what do I do?

Answer 4

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?

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

Is that the answer?

Answer 9

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

Answer 10

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

Answer 11

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:
\[\Large {A_1},\;{A_2},\;{A_3},\;{A_4},\;{A_5}\]

Answer 12

which*

Answer 13

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

Answer 14

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"

Answer 15

Thank-you:)

Answer 16

:)

Answer 17

ok!

Answer 18

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

Answer 19

I agree

Answer 20

here is the corresponding graph:

Answer 21

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

Answer 22

yes! I think so!

Answer 23

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"

Answer 24

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

Answer 25

:)

Answer 26

Sorry, I have one last question!

Answer 27

I'm pondering...

Answer 28

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

Answer 29

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