Question

graph the polynomial function f(x)=(x - 1)(x + 4)^2 using parts (a) through (e). (a) determine the end - behavior of the graph of the function. the graph of f behaves like y = for large values of |x|.

Explanation:

Step1: Expand the polynomial

First, expand \((x + 4)^2=x^{2}+8x + 16\). Then \(f(x)=(x - 1)(x^{2}+8x + 16)=x^{3}+8x^{2}+16x-x^{2}-8x - 16=x^{3}+7x^{2}+8x - 16\).

Step2: Analyze end - behavior

For a polynomial function \(y = a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_0\), the end - behavior is determined by the leading term \(a_nx^n\). Here, \(n = 3\) (odd) and \(a_n=1\) (positive). For large \(|x|\), when \(x\to+\infty\), \(y\to+\infty\) and when \(x\to-\infty\), \(y\to-\infty\). The graph behaves like \(y = x^{3}\) for large values of \(|x|\).

Answer:

\(x^{3}\)