Question

graph the polynomial function f(x)=x(3 - x)(7 - x) 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

\[

$$\begin{align*} f(x)&=x(3 - x)(7 - x)\\ &=x(21-3x - 7x+x^{2})\\ &=x(x^{2}-10x + 21)\\ &=x^{3}-10x^{2}+21x \end{align*}$$

\]

Step2: Analyze end - behavior

For a polynomial function \(y = a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0\), the end - behavior is determined by the leading term \(a_nx^n\). Here, \(n = 3\) (odd) and \(a_n=1\) (positive). For large values of \(|x|\), the function behaves like the leading term.

Answer:

\(y = x^{3}\)