Before finding the function that describes the graph we can at a glance identify the following:
Y-intercepts
[tex]\begin{gathered} (0,4) \\ y=4\to x=0 \end{gathered}[/tex]X-intercepts or the roots
[tex]\begin{gathered} x=-1 \\ x=1 \\ x=2 \end{gathered}[/tex]Other point given
[tex](3,16)[/tex]with the given points we will replace in each function to identify the correct one. we will start with (0,4)
[tex]\begin{gathered} a) \\ f(0)=3(0-2)(0+1)^2 \\ f(0)=3(-2)(1) \\ f(0)=-6_{}\to\text{Not correct} \end{gathered}[/tex][tex]\begin{gathered} b) \\ f\mleft(0\mright)=(0-2)\mleft(0+1\mright)\mleft(0-3\mright) \\ f(0)=(-2)(1)(-3) \\ f(0)=6\to\text{Not correct} \end{gathered}[/tex][tex]\begin{gathered} c) \\ f\mleft(0\mright)=2\mleft(0-1\mright)\mleft(0+1\mright)\mleft(0-2\mright) \\ f(0)=2(-1)(1)(-2) \\ f(0)=4\to is\text{ Possible} \end{gathered}[/tex][tex]\begin{gathered} d) \\ f\mleft(0\mright)=4\mleft(0-3\mright)\mleft(0-2\mright)\mleft(0+1\mright) \\ f(0)=4(-3)(-2)(1) \\ f(0)=24\to\text{Not correct} \end{gathered}[/tex]After analyzing the y-intercept we can see that the function it fulfills is letter C
To verify it, we will replace the additional point and see if it gives the expected result:
[tex]\begin{gathered} f(3)=2(3-1)(3+1)(3-2) \\ f(3)=2(2)(4)(1) \\ f(3)=16\to\text{Correct} \end{gathered}[/tex]With this we can confirm that the function of the letter C describes the figure
