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 = - 3,0,3 (type an integer or a decimal. use a comma to separate answers as needed.)
at which zeros does the graph of the function cross the x - axis? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. x = - 3,3 (type an integer or a decimal. use a comma to separate answers as needed.)
b. there are no x - intercepts at which the graph crosses the x - axis.
at which zeros does the graph of the function touch the x - axis and turn around? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. x = (type an integer or a decimal. use a comma to separate answers as needed.)
b. there are no x - intercepts at which the graph touches the x - axis and turns around.

Explanation:

Step1: Analyze leading - coefficient and degree

The function $f(x)=x^{4}-9x^{2}$ is a polynomial function. The degree $n = 4$ (even) and the leading coefficient $a = 1$ (positive). According to the leading - coefficient test, for a polynomial function $y = a_nx^n+\cdots+a_0$, when $n$ is even and $a_n>0$, the graph rises left and rises right.

Step2: Find x - intercepts

Set $f(x)=0$, so $x^{4}-9x^{2}=0$. Factor out $x^{2}$: $x^{2}(x^{2}-9)=0$. Then factor $x^{2}-9$ as a difference of squares: $x^{2}(x - 3)(x + 3)=0$. Using the zero - product property, $x^{2}=0$ gives $x = 0$, $x-3=0$ gives $x = 3$, and $x + 3=0$ gives $x=-3$.

Step3: Determine crossing and touching of x - axis

For a polynomial function $y = f(x)$ and a zero $x = c$, if the factor corresponding to $c$ has an odd multiplicity, the graph crosses the $x$-axis at $x = c$, and if the factor has an even multiplicity, the graph touches the $x$-axis and turns around at $x = c$.
The factor $x$ in $f(x)=x^{2}(x - 3)(x + 3)$ has multiplicity 2, and the factors $(x - 3)$ and $(x + 3)$ have multiplicity 1. So the graph crosses the $x$-axis at $x=-3$ and $x = 3$, and touches the $x$-axis and turns around at $x = 0$.

Answer:

a) A. The graph of f(x) rises left and rises right.
b) $x=-3,0,3$; At which zeros does the graph of the function cross the x - axis? A. $x=-3,3$; At which zeros does the graph of the function touch the x - axis and turn around? A. $x = 0$