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 to be used in your explanations.

Answer 1

First. Finding the x-intercepts of
`W(x)`

Let
`W(x)` be the change in water level. So to find the x-intercepts of this function we can use The Rational Zero Test that states:

To find the zeros of the polynomial:


`f(x)=a_(n)x^(n)+a_(n-1)x^(n-1)+...+a_(2)x^(2)+a_(1)x+a_(0)`

We use the Trial-and-Error Method which states that a factor of the constant term:


`a_(0)`

can be a zero of a polynomial (the x-intercepts).

So let's use an example: Suppose you have the following polynomial:


`W(x)=x^(4)-x^(3)-7x^(2)+x+6`

where the constant term is
`a_(0)=6`. The possible zeros are the factors of this term, that is:


`1, -1, 2, -2, 3, -3, 6 \ and \ -6`.

Thus:


`W(1)=0 \\ W(-1)=0 \\ W(2)=-12 \\ W(-2)=0 \\ W(3)=0 \\ W(-3)=48 \\ W(6)=840 \\ W(-6)=1260`

From the foregoing, we can affirm that
`1, -1, -2 \ and \ 3` are zeros of the polynomial.

Second. Construction a rough graph of
`W(x)`

Given that this is a polynomial, then the function is continuous. To graph it we set the roots on the coordinate system. We take the interval:


`[-2,-1]`

and compute
`W(c)` where
`c` is a real number between -2 and -1. If
`W(c)>0`, the curve start rising, if not, the curve start falling. For instance:


`If \ c=-(3)/(2) \\ \\ then \ w(-(3)/(2))=-2.81`

Therefore the curve start falling and it goes up and down until
`x=3` and from this point it rises without a bound as shown in the figure below