The most appropriate polynomial function for this graph would be [tex]\frac{1}{2} (x-4)(x+1)^{2}[/tex].
A polynomial function is a function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation, etc.
To write a polynomial function for the given graph -
First, determine the end behavior of the graph.
As visible,
As x tends to ∞, f(x) tends to ∞.
As x tends to -∞, f(x) tends to -∞.
This implies that the polynomial has an odd degree, eg. 3,5 etc.
To determine the degree of the polynomial, we check the turning points. The graph has 2 turning points.
We can say that, if a graph has n degree, it has at most n-1 turning points. Using this we say that the function has degree 3.
Now, using the graph, we mark the x intercepts.
(4,0) is first intercept. We check the nature of the graph at that point. It is a straight line. Thus (4,0) is a linear solution in the function. Let's say (x-4).
(-1,0) is the second intercept. We check the nature of the graph at that point. It is a polynomial. Thus (-1,0) is a quadratic solution in the function. Let's say [tex](x+1)^{2}[/tex].
Tentative polynomial function becomes f(x) = [tex]a(x-4)(x+1)^{2}[/tex].
We see that (0,-2) is the y intercept in graph. It must satisfy our equation. Solving that we find that a = 1/2.
Thus, the given graph is for polynomial function f(x) = [tex]\frac{1}{2} (x-4)(x+1)^{2}[/tex]
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