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:

$If \ c=-\frac{3}{2} \\ \\ then \ w(-\frac{3}{2})=-2.81$

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

Answer 2

Suppose a polynomial function which represents water level in the lake is given by:

W(x)=(x-1) (Ax-B)(C x -D)(P x -Q)( x -S), which is polynomial function of degree 5, showing water level on five different days.

As, temperature will vary on different days, so water level will change in the lake at different days of month or year.

X intercepts can be found by,

Put , W(x) =0

(x-1) (Ax-B)(C x -D)(P x -Q)( x -S)=0

→x-1=0,∧ Ax-B=0,∧C x -D=0,∧P x-Q=0, ∧ x-S=0

→x=1, $x=\frac{B}{A}, x=\frac{D}{C}, x=\frac{Q}{P}, x=S\\\\ 1$

These are x intercepts.That is water level on five different days.

There are chances that water level in lake increase if rain happens, considering that there is no cats and dogs in between the period so, these are representation of water in lake on on five different days.

On, first day, that is ,when, x=1, there will be no change in water level in the lake.

As, graph of polynomial function is like this, but x intercepts will be just  lines , x=1, x=B/A,... passing through these five points.