Question

describe the end behavior of the following function: f(x) = 2x⁴ + x³ a. the graph of the function starts low and ends low. b. the graph of the function starts high and ends high. c. the graph of the function starts high and ends low. d. the graph of the function starts low and ends high.

Explanation:

Step1: Identify the leading term

The leading term of the polynomial \( F(x) = 2x^4 + x^3 \) is \( 2x^4 \) (the term with the highest degree).

Step2: Analyze the degree and leading coefficient

• The degree of the leading term (\( 2x^4 \)) is 4, which is even.
• The leading coefficient is 2, which is positive.

For a polynomial function, the end - behavior is determined by the leading term:

• If the degree is even and the leading coefficient is positive, as \( x

ightarrow+\infty \), \( y = a_nx^n+\cdots
ightarrow+\infty \) (because \( x^n \) is positive when \( n \) is even and \( x
ightarrow+\infty \), and multiplying by a positive \( a_n \) keeps it positive and large).

• As \( x

ightarrow-\infty \), since \( n = 4 \) (even), \( x^4=(-x)^4>0 \), and multiplying by a positive leading coefficient \( a_n = 2 \), we have \( y = a_nx^n+\cdots
ightarrow+\infty \).

This means that as \( x \) approaches \( -\infty \) (the left - hand end) and \( x \) approaches \( +\infty \) (the right - hand end), the function values go to \( +\infty \). So the graph starts high (when \( x
ightarrow-\infty \)) and ends high (when \( x
ightarrow+\infty \)).

Answer:

B. The graph of the function starts high and ends high.