Question

Let the graph of f(x) be given below. Find the formula of f(x), a polynomial function, of least degree.

Answer 1

`f(x)\text{ = }(1)/(2)x(x\text{ }+2)(x-4)`

Step-by-step explanation:

To detrmine the formula of the polynomial, we check for the roots on the graph:

when y = 0, x = -2

when y = 0, x = 4

We have two roots.

x = -2

x + 2 = 0

x = 4

x - 4 = 0

3rd factor is x = 0

Hence, we have two factors: x(x + 2) and (x - 4)

The polynomial function using the factors:

`f(x)\text{ = ax(x + 2)(x - 4)}`

Next, we find the value of a:

To get a , we pick a point on the graph. let the point be (0, -4)

substitute the point in the function above:

`\begin{gathered} f(x)\text{ = y = -4, x = 0} \\ -4\text{ = a(0 + 2) (0 - 4)} \\ -4\text{ = a(2)(-4)} \\ -4\text{ = -8a} \\ a\text{ = -4/-8} \\ a\text{ = 1/2} \end{gathered}`

The formula of the polynomial becomes:

`f(x)\text{ = }(1)/(2)x(x\text{ }+2)(x-4)`