Answer: Graph D will be correct graph for the given function.
Explanation:
Given function \(f(x) = x^4-2x^3-3x^2+4x+1\)
Since it is a bi-quadratic equation thus it must have 4 roots and (0,1) is one of its point.
Moreover, the degree of the function is even thus the end behavior of the function is\(f(x)\to+\infty\), as \(x\to-\infty\) and \(f(x)\to+\infty\) as \(x\to+\infty\)
In graph A, function has four root but it does not have the end behavior same as function f(x).( because in this graph \(f(x)\to-\infty\), as \(x\to-\infty\) and \(f(x)\to-\infty\), as \(x\to+\infty\).) so, it can not be the graph of given function.
In graph B, neither it has four root nor it has the end behavior same as function f(x).(because in this graph \(f(x)\to+\infty\) as \(x\to-\infty\) and \(f(x)\to-\infty\) as \(x\to+\infty\).) so, it can not be the graph of given function.
In graph C, neither it has four root nor it has the same end behavior as function f(x).(because in this graph \(f(x)\to-\infty\) as \(x\to-\infty\) and \(f(x)\to+\infty\) as\(x\to\infty\).) so, it also can not be the graph of given function.
In graph D it has four root as well as it has the same end behavior as the given function. Also it passes through the point (0,1).
Thus, graph D is the graph of given function.
