Answer:
[tex]\displaystyle f(x)=-\frac{2}{3}(x+2)(x-3)^2[/tex]
Step-by-step explanation:
The graph corresponds to a cubic function of the form:
[tex]f(x)=a(x-p)(x-q)(x-r)[/tex]
Where p, q, and r are the zeros of f(x).
We can clearly see there are only two crossings through the x-axis. That is because one of the roots is repeated (multiple).
Thus, the roots are p=-2, q=r=3
Substituting into the function:
[tex]f(x)=a(x+2)(x-3)(x-3)[/tex]
[tex]f(x)=a(x+2)(x-3)^2[/tex]
The value of a can be found by using the y-intercept seen on the graph (0,-12):
[tex]-12=a(0+2)(0+3)^2[/tex]
Operating:
[tex]-12=18a[/tex]
Thus:
[tex]a = -12 / 18 = -2/3[/tex]
The function is now complete:
[tex]\mathbf{\displaystyle f(x)=-\frac{2}{3}(x+2)(x-3)^2}[/tex]
