Question

In Problems 65–68, construct a polynomial function that might have the given graph. (More than one answer may be possible.)

Short answer

The possible function is $f\left(x\right)=-\frac{1}{2}\left(x+1\right)\left(x-1\right)\left(x-2\right)$

Step-by-step solution

Step 1. Given information   

The graph is

Step 2. To find the possible function from graph.   

In the given graph we can see that there are 3 zeros because the graph is crossing the x-axis at

$x=-1,1,2$, so these zeros have odd multiplicity

So the factors of the function are:

$\left(x+1\right),\left(x-1\right),\left(x-2\right)$

the function is

role="math" localid="1646322635488" $f\left(x\right)=a\left(x+1\right)\left(x-1\right)\left(x-2\right)$

Step 3. To find value of a

Since,$f\left(0\right)=-1$

hence

$$\begin{gathered} -1=a(0+1)(0-1)(0-2) \\ -1=a\left(1\right)(-1)(-2) \\ -1=2a \\ a=-\frac{1}{2} \end{gathered}$$

so the function is

$f\left(x\right)=-\frac{1}{2}\left(x+1\right)\left(x-1\right)\left(x-2\right)$