Answer :
The graph of a function is given. It is required to find the function that represents the graph.
Notice from the graph that the function has roots or zeros at -3,-2, and 2.
Hence, the function takes the form:
[tex]\begin{gathered} f(x)=a(x-(-3))(x-(-2))(x-2) \\ \Rightarrow f(x)=a(x+3)(x+2)(x-2) \end{gathered}[/tex]Where a is a constant to be determined.
Notice that the given graph passes through the point (0,-12).
Substitute x=0 and f(x)=-12 into the equation of the function to find a:
[tex]\begin{gathered} -12=a(0+3)(0+2)(0-2) \\ \Rightarrow-12=a(3)(2)(-2) \\ \Rightarrow-12=-12a \\ \text{ Swap the sides of the equation:} \\ \Rightarrow-12a=-12 \\ \text{ Divide both sides by }-12: \\ \Rightarrow\frac{-12a}{-12}=\frac{-12}{-12} \\ \Rightarrow a=1 \end{gathered}[/tex]Hence, the function is:
[tex]\begin{gathered} f(x)=1(x+3)(x+2)(x-2) \\ \Rightarrow f(x)=(x+2)(x+3)(x-2) \end{gathered}[/tex]