Question

sketch the polynomial function using the following step - by - step process.
f(x)=x^{2}(x - 6)
a. determine the end behavior of the graph of f.
the left - hand behavior starts and the right - hand behavior ends

Explanation:

Step1: Expand the polynomial

$$f(x)=x^{2}(x - 6)=x^{3}-6x^{2}$$
The leading term is \(x^{3}\) (degree \(n = 3\), which is odd, and leading coefficient \(a=1>0\)).

Step2: Determine end - behavior rules

For a polynomial \(y = a_nx^{n}+a_{n - 1}x^{n-1}+\cdots+a_0\) with \(n\) (degree) odd and \(a_n>0\):
As \(x\to-\infty\), \(y\to-\infty\) (left - hand behavior).
As \(x\to+\infty\), \(y\to+\infty\) (right - hand behavior).

Answer:

The left - hand behavior starts at \(-\infty\) and the right - hand behavior ends at \(+\infty\)