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

a) Use the Leading Coefficient Test to determine the graph’s end beha
A. The graph of falls left and rises right.
B. The graph of rises left and rises right.
C. The graph of falls left and falls right.
D. The graph of rises left and falls right.
b) Find the -intercepts.

(Type an integer or a decimal. Use a comma to separate answers as

Ask by Lyons Harper. in the United States

Mar 29,2025

Question

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

a) Use the Leading Coefficient Test to determine the graph’s end beha
A. The graph of falls left and rises right.
B. The graph of rises left and rises right.
C. The graph of falls left and falls right.
D. The graph of rises left and falls right.
b) Find the -intercepts.

(Type an integer or a decimal. Use a comma to separate answers as

Ask by Lyons Harper. in the United States

Mar 29,2025

UpStudy AI · Accepted Answer

**(a)**  
The leading term of $$ f(x)=x^4-x^2 $$ is $$ x^4 $$. The degree is 4 (an even number) and the leading coefficient is positive. By the Leading Coefficient Test, as $$ x \to -\infty $$ and $$ x \to \infty $$, $$ f(x) \to \infty $$.

Thus, the correct answer is:  
$$ \textbf{B. The graph of } f(x) \textbf{ rises left and rises right.} $$

---

**(b)**  
To find the $$ x $$-intercepts, set $$ f(x)=0 $$:

$$
x^4-x^2=0
$$

Factor out $$ x^2 $$:

$$
x^2 (x^2-1)=0
$$

Recognize that $$ x^2-1 $$ can be factored as a difference of squares:

$$
x^2 (x+1)(x-1)=0
$$

Set each factor equal to zero:

1. $$ x^2=0 $$ leads to:
   $$
   x=0
   $$
2. $$ x+1=0 $$ leads to:
   $$
   x=-1
   $$
3. $$ x-1=0 $$ leads to:
   $$
   x=1
   $$

Thus, the $$ x $$-intercepts are:

$$
x=-1, \, 0, \, 1
$$
**(a)** The graph of $$ f(x) $$ rises left and rises right. **(b)** The $$ x $$-intercepts are $$ x=-1, 0, 1 $$.

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