Question

use the leading - coefficient test to determine the end behavior of the graph of the given polynomial function. then use this end behavior to match the polynomial function with its graph.
f(x)=(x + 2)^6
choose the correct graph below.

Explanation:

Step1: Expand the polynomial

Using the binomial theorem \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\), for \(f(x)=(x + 2)^6\), the leading - term is \(x^6\) (since when we expand \((x+2)^6=\binom{6}{0}x^{6}2^{0}+\cdots\)). The leading coefficient is \(1\) (the coefficient of \(x^6\)) and the degree \(n = 6\) (an even number).

Step2: Determine end - behavior

For a polynomial function \(y = a_nx^n+\cdots+a_0\) with \(a_n
eq0\), when \(n\) is even and \(a_n>0\), as \(x\to-\infty\), \(y\to+\infty\) and as \(x\to+\infty\), \(y\to+\infty\).

Step3: Analyze the zero of the function

Set \(f(x)=(x + 2)^6=0\), we get \(x=-2\) as a zero of multiplicity \(6\). A zero of even multiplicity means the graph touches the \(x\) - axis at that point.

Answer:

B.