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

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

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

and touches x-axis at $x=1$

so it has even multiplicity.

So the factors of the function are:

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

the function is $y=a\left(x+1\right){\left(x-1\right)}^2\left(x-2\right)$

Step 3. To find value of a

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

hence

role="math" localid="1646321911044" $$\begin{gathered} 1=a(0+1){(0-1)}^2(0-2) \\ 1=a\left(1\right){(-1)}^2(-2) \\ 1=-2a \\ a=-\frac{1}{2} \end{gathered}$$

so the function is

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