Question

Find a polynomial function f(x) at least possible degree having the graph shown.

Answer 1

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

`f(x)=a(x+2)(x-3)^2`

we are left with finding a

notice the coordinate on the y-axis is (0,9)

substituting,

`\begin{gathered} f(x)=a(x+2)(x-3)^2 \\ f(0)=a(2)(-3)^2 \\ 9=18a \\ a=(9)/(18) \\ a=(1)/(2) \end{gathered}`

Therefore,

`f(x)=(1)/(2)(x+2)(x-3)^2`