Determining a Polynomial Function from Its Graph

It is possible to write a polynomial function from its graph if the zeros and any other point on the graph are known. Suppose that you are asked to find a polynomial function of least possible degree having the graph shown below.

A comprehensive graph of the function.PNG

Because the graph crosses the x-axis at 1 and 3 and bounces at -2, we know that the factored form of the function is $LaTeX: f\left(x\right)=a\left(x-1\right)\left(x-3\right)\left(x+2\right)^2$ $f\left(x\right)=a\left(x-1\right)\left(x-3\right)\left(x+2\right)^2$ .

Think of it this way: The same function with the exponents representing the multiplicities pointed out.PNG

Now find the value of a by substituting the x- and y-values of any other point on the graph, say (0,−12), into this function and solving for a.

$LaTeX: f\left(x\right)=a\left(x-1\right)\left(x-3\right)\left(x+2\right)^2$ $f\left(x\right)=a\left(x-1\right)\left(x-3\right)\left(x+2\right)^2$

$LaTeX: -12=a\left(0-1\right)\left(0-3\right)\left(0+2\right)^2$ $-12=a\left(0-1\right)\left(0-3\right)\left(0+2\right)^2$

$LaTeX: -12=a\left(12\right)$ $-12=a\left(12\right)$

LaTeX: a=-1 $a=-1$

So, the graph is of the function $LaTeX: f\left(x\right)=-\left(x-1\right)\left(x-3\right)\left(x+2\right)^2$ $f\left(x\right)=-\left(x-1\right)\left(x-3\right)\left(x+2\right)^2$

(Of course you may already know this just by recognizing it is the same graph from the previous page!)