Question

answer parts a.- e. for the function shown below.
f(x)=3x^{2}+x^{3}
a. use the leading - coefficient test to determine the graphs end behavior. which statement describes the end behavior of f(x)?
a. the graph of f(x) rises left and falls right.
b. the graph of f(x) rises left and rises right.
c. the graph of f(x) falls left and rises right.
d. the graph of f(x) falls left and falls right.

Explanation:

Step1: Identify the degree and leading coefficient

The function is \(f(x)=x^{3}+3x^{2}\), the degree \(n = 3\) (odd) and the leading - coefficient \(a = 1\) (positive).

Step2: Apply the leading - coefficient test

For a polynomial function \(y = a_nx^n+\cdots+a_0\), when \(n\) is odd and \(a_n>0\), as \(x\to-\infty\), \(y\to-\infty\) (the graph falls to the left) and as \(x\to+\infty\), \(y\to+\infty\) (the graph rises to the right).

Answer:

C. The graph of \(f(x)\) falls left and rises right.