Question

You are having a conference call with theCEO of a paper company. You have interrupted the number of trees cut down verses profit as the function p(x)=-x4+x3+7x2-x-6. Describe to the CEO what the graph looks like and in general how to sketch the graph without using technology. Use complete sentences and focus on the end behaviors of the graph and where the company will break even when p(x)=0.

Answer 1

You can answer this by first attempting to simplify the polynomial to find the roots. Here I attempt to simplify it by using division.

$p(x)=-x^{4}+x^{3}+7x^{2}-x-6$
$p(x)=(x-3)(-x^{3}-2x^{2}+x+2)$
$p(x)=(x-3)(x+2)(-x^{2}+1)$

Now let us try setting p(x)=0 to compute for where the company will break even.

$p(x)=(x-3)(x+2)(-x^{2}+1)$

$x_{1} -3=0$
$x_{1}=3$

$x_{2} +2=0$
$x_{2} =-2$

$- x_{3} ^{2}+1=0$
$x_{3} ^{2}=1$
$x_{3} =1$ or $x_{3} =-1$

Now we have four roots, of which none is repeating so therefore we know that the graph will pass by the x-axis four times. The first term of the polynomial is negative and the degree is positive so the graph will start as an increasing line and it will end decreasing.

We can also quickly see that the y-intercept is (0,-6) since it is the only constant term in the function.

Knowing all these, you can tell the CEO that the graph will start at the bottom, pass by (-2,0), increase then decrease until it passes (-1,0) to which it will continue decreasing until it passes (0,-6) where it will increase until it crosses the x-axis again at (1,0) and then finally make one last increase and decrease until it passes (3,0) and then continue to decrease forever.

Since there are no negative trees to cut, you know that the company will break even when the number of trees cut is equal to 1 or 3.