In order to determine the form of the function of the graph, take into account that the given graph belongs to a cubic polynomial.
The zeros of the curve helps to determine the factors of the polynomial.
As you can notice, the are two zeros on the graph, for x = and x = 3. It suggest that the factors of the polynomial are (x - 3)x. However, due to the graph is of third order, it is necessary that one of the factors has multiplicity 2. If x is such the factor, you have for f(x):
f(x) = x^2(x - 3)
Now, take into account that the orientation of the curve demands that the previous factors are preceded by -1, then:
f(x) = -x^2(x - 3)
If you graph the previous expression, you obtain:
The graph does not macht with the given graph yet.
In order to the previous graph equal the given graph, you can notice that it is necessary to move the curve 4 units down and 1 unit to the right. It can be done by subtracting 4 to the complete expression, and furthermore, by adding -1 to x into the expression for f(x) we have.
Then, now we have:
f(x) = -(x-1)^2*(x - 1 - 3) - 4
f(x) = -(x - 1)^2(x - 4) - 4
When you graph the previous function, you get:
Hence, the function is:
f(x) = -(x - 1)^2(x - 4) - 4