Question

use the given function to complete parts a) through e) below.
f(x) = -x⁴ + 16x²

a) use the leading coefficient test to determine the graphs end behavior.
○ a. the graph of f(x) rises left and falls right.
○ b. the graph of f(x) falls left and falls right.
○ c. the graph of f(x) rises left and rises right.
○ d. the graph of f(x) falls left and rises right.

Explanation:

Step1: Identify Degree and Leading Coefficient

The function is \( f(x) = -x^4 + 16x^2 \). The degree (highest power of \( x \)) is 4 (even), and the leading coefficient (coefficient of the highest - degree term) is - 1 (negative).

Step2: Apply Leading Coefficient Test

For a polynomial function \( f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0 \):

• If the degree \( n \) is even:
• If the leading coefficient \( a_n>0 \), the graph rises to the left and rises to the right.
• If the leading coefficient \( a_n < 0 \), the graph falls to the left and falls to the right.

Since our polynomial has an even degree (\( n = 4 \)) and a negative leading coefficient (\( a_4=-1<0 \)), the graph of \( f(x) \) falls left and falls right.

Answer:

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