Given the graph
We want to find the polynomial of the graph f(x)
Solution
From the graph, one can tell that the its is a polynomial of power three
we first notice that the graph crosses the x-axis at x=-2 and x=3
so, x=-2 and x=3 are the roots of f(x)
Notice that at the root x=3, the graph change direction
That implies that x=3 (twice)
so we have
[tex]f(x)=a(x+2)(x-3)^2[/tex]
we are left with finding a
notice the coordinate on the y-axis is (0,9)
substituting,
[tex]\begin{gathered} f(x)=a(x+2)(x-3)^2 \\ f(0)=a(2)(-3)^2 \\ 9=18a \\ a=\frac{9}{18} \\ a=\frac{1}{2} \end{gathered}[/tex]
Therefore,
[tex]f(x)=\frac{1}{2}(x+2)(x-3)^2[/tex]
