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.
b) find the x - intercepts.
x =
(type an integer or a decimal. use a comma to separate answers as needed.)

Explanation:

Step1: Identify the leading - coefficient and degree

The function is $f(x)=x^{4}-9x^{2}$, the degree $n = 4$ (even) and the leading - coefficient $a = 1$ (positive). According to the leading - coefficient test, when $n$ is even and $a>0$, the graph rises to the left and rises to the right.

Step2: Find the x - intercepts

Set $f(x)=0$, so $x^{4}-9x^{2}=0$. Factor out $x^{2}$: $x^{2}(x^{2}-9)=0$. Then, use the difference - of - squares formula $x^{2}-9=(x + 3)(x - 3)$. So we have $x^{2}(x + 3)(x - 3)=0$.
Set each factor equal to zero:

• $x^{2}=0$ gives $x = 0$.
• $x+3=0$ gives $x=-3$.
• $x - 3=0$ gives $x = 3$.

Answer:

a) A. The graph of f(x) rises left and rises right.
b) $x=-3,0,3$