Question

use the given function to complete parts (a) through (e) below.

f(x)=x^4 - 9x^2

a) use the leading coefficient test to determine the graphs end behavior.

a. the graph of f(x) rises left and rises right.
b. the graph of f(x) falls left and falls right.
c. the graph of f(x) falls left and rises right.
d. the graph of f(x) rises left and falls right.

Explanation:

Step1: Identify the leading - term

The function is \(f(x)=x^{4}-9x^{2}\), and the leading - term is \(x^{4}\) (the term with the highest power of \(x\)).

Step2: Determine the degree and leading coefficient

The degree \(n = 4\) (even) and the leading coefficient \(a = 1\) (positive).

Step3: Apply the Leading - Coefficient Test

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

Answer:

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