to understand this graphs you must find the roots on each of the functions.
start by funtion 1.
[tex]\begin{gathered} x^3+3x^2=0 \\ x\cdot(x^2+3x)=0 \\ x=0 \\ (x^2+3x)=0 \\ x(x+3)=0 \\ x=0 \\ x+3=0 \\ x=-3 \end{gathered}[/tex]for function 1 you will need to find a graph that only intercept the x-axis on 0 an -3. In this case it will be the graph A.
Do the same for each function
[tex]\begin{gathered} -x\cdot(x-1)\cdot(x+2) \\ x=0 \\ x-1=0 \\ x=1 \\ x+2=0 \\ x=-2 \end{gathered}[/tex]function 2, the interceptions are 0,1 and -2. Graph C will be the correct one for this function
function 3
[tex]\begin{gathered} -x^3+3x^2=0 \\ x\cdot(-x^2+3x)=0 \\ x=0 \\ (-x^2+3x)=0 \\ x(-x+3)=0 \\ x=0 \\ -x+3=0 \\ x=3 \end{gathered}[/tex]for fuction 3, roots will be 0 and 3, the associated graph will be D
and lastly the roots for function 4.
[tex]\begin{gathered} -x\cdot(x+1)\cdot(x-2) \\ x=0 \\ x+1=0 \\ x=-1 \\ x-2=0 \\ x=2 \end{gathered}[/tex]The associated graph is B.
